Properties

Label 8-507e4-1.1-c3e4-0-6
Degree $8$
Conductor $66074188401$
Sign $1$
Analytic cond. $800748.$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s + 2·4-s + 90·9-s − 24·12-s − 125·16-s − 144·17-s − 276·23-s − 310·25-s − 540·27-s + 12·29-s + 180·36-s + 940·43-s + 1.50e3·48-s − 340·49-s + 1.72e3·51-s − 2.26e3·53-s + 320·61-s − 380·64-s − 288·68-s + 3.31e3·69-s + 3.72e3·75-s + 8·79-s + 2.83e3·81-s − 144·87-s − 552·92-s − 620·100-s + 636·101-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/4·4-s + 10/3·9-s − 0.577·12-s − 1.95·16-s − 2.05·17-s − 2.50·23-s − 2.47·25-s − 3.84·27-s + 0.0768·29-s + 5/6·36-s + 3.33·43-s + 4.51·48-s − 0.991·49-s + 4.74·51-s − 5.87·53-s + 0.671·61-s − 0.742·64-s − 0.513·68-s + 5.77·69-s + 5.72·75-s + 0.0113·79-s + 35/9·81-s − 0.177·87-s − 0.625·92-s − 0.619·100-s + 0.626·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(800748.\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + p T )^{4} \)
13 \( 1 \)
good2$C_2^2$ \( ( 1 - T^{2} + p^{6} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 + 62 p T^{2} + 47931 T^{4} + 62 p^{7} T^{6} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 + 340 T^{2} + 858 p^{2} T^{4} + 340 p^{6} T^{6} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 2644 T^{2} + 5200842 T^{4} + 2644 p^{6} T^{6} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 + 72 T + 11071 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 16468 T^{2} + 139269162 T^{4} + 16468 p^{6} T^{6} + p^{12} T^{8} \)
23$D_{4}$ \( ( 1 + 6 p T + 26596 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 6 T + 24103 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 22684 T^{2} - 285715578 T^{4} + 22684 p^{6} T^{6} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 93418 T^{2} + 4535499099 T^{4} + 93418 p^{6} T^{6} + p^{12} T^{8} \)
41$D_4\times C_2$ \( 1 + 270250 T^{2} + 27758790507 T^{4} + 270250 p^{6} T^{6} + p^{12} T^{8} \)
43$D_{4}$ \( ( 1 - 470 T + 202764 T^{2} - 470 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 129172 T^{2} + 10043128410 T^{4} + 129172 p^{6} T^{6} + p^{12} T^{8} \)
53$D_{4}$ \( ( 1 + 1134 T + 602719 T^{2} + 1134 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 41740 T^{2} + 83104566582 T^{4} + 41740 p^{6} T^{6} + p^{12} T^{8} \)
61$D_{4}$ \( ( 1 - 160 T + 74343 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 1146724 T^{2} + 509040985386 T^{4} + 1146724 p^{6} T^{6} + p^{12} T^{8} \)
71$D_4\times C_2$ \( 1 + 1134052 T^{2} + 574891463802 T^{4} + 1134052 p^{6} T^{6} + p^{12} T^{8} \)
73$C_2^2$ \( ( 1 + 626159 T^{2} + p^{6} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 4 T + 176406 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 124420 T^{2} + 286548422442 T^{4} + 124420 p^{6} T^{6} + p^{12} T^{8} \)
89$D_4\times C_2$ \( 1 + 818116 T^{2} + 1140216422502 T^{4} + 818116 p^{6} T^{6} + p^{12} T^{8} \)
97$D_4\times C_2$ \( 1 - 1862660 T^{2} + 2429934389382 T^{4} - 1862660 p^{6} T^{6} + p^{12} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.80682199825930562728990412766, −7.71623025900656791641355529437, −7.38172216807519640360987604748, −7.18865853427112873591183271993, −6.58592266294271919501154111480, −6.56719344942428786780915256712, −6.44052118598353183017102799416, −6.39703673118474459700452791099, −5.93187246775876670481689550050, −5.90142407623043661802220153470, −5.52317212175414376291587927734, −5.19297283451489861832987095013, −5.03299782136130514998042208613, −4.50493939340056767701065161529, −4.39872582696397137206031941712, −4.24676278489650768344372372894, −4.20841270190841640600250909533, −3.54595320269453940479262184079, −3.51448748757758343868187459525, −2.55472301057076868574984331212, −2.51229610515168250121098752382, −2.02672685350344045211732700666, −1.99114258243882634064201701255, −1.27269060929327938250794938671, −1.26249770485836946364718737188, 0, 0, 0, 0, 1.26249770485836946364718737188, 1.27269060929327938250794938671, 1.99114258243882634064201701255, 2.02672685350344045211732700666, 2.51229610515168250121098752382, 2.55472301057076868574984331212, 3.51448748757758343868187459525, 3.54595320269453940479262184079, 4.20841270190841640600250909533, 4.24676278489650768344372372894, 4.39872582696397137206031941712, 4.50493939340056767701065161529, 5.03299782136130514998042208613, 5.19297283451489861832987095013, 5.52317212175414376291587927734, 5.90142407623043661802220153470, 5.93187246775876670481689550050, 6.39703673118474459700452791099, 6.44052118598353183017102799416, 6.56719344942428786780915256712, 6.58592266294271919501154111480, 7.18865853427112873591183271993, 7.38172216807519640360987604748, 7.71623025900656791641355529437, 7.80682199825930562728990412766

Graph of the $Z$-function along the critical line