Properties

Label 4-3696e2-1.1-c1e2-0-6
Degree $4$
Conductor $13660416$
Sign $1$
Analytic cond. $870.999$
Root an. cond. $5.43256$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 6·5-s − 2·7-s + 3·9-s + 2·11-s + 2·13-s + 12·15-s + 6·17-s + 4·19-s − 4·21-s + 2·23-s + 17·25-s + 4·27-s + 2·29-s + 2·31-s + 4·33-s − 12·35-s + 2·37-s + 4·39-s − 4·41-s + 6·43-s + 18·45-s − 12·47-s + 3·49-s + 12·51-s − 10·53-s + 12·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 2.68·5-s − 0.755·7-s + 9-s + 0.603·11-s + 0.554·13-s + 3.09·15-s + 1.45·17-s + 0.917·19-s − 0.872·21-s + 0.417·23-s + 17/5·25-s + 0.769·27-s + 0.371·29-s + 0.359·31-s + 0.696·33-s − 2.02·35-s + 0.328·37-s + 0.640·39-s − 0.624·41-s + 0.914·43-s + 2.68·45-s − 1.75·47-s + 3/7·49-s + 1.68·51-s − 1.37·53-s + 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13660416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13660416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13660416\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(870.999\)
Root analytic conductor: \(5.43256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3696} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13660416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.62923707\)
\(L(\frac12)\) \(\approx\) \(10.62923707\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 97 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 129 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.721218405147397565040916811109, −8.500689446833055001099449387147, −8.042568480195725916233526167866, −7.67137938126613580363559479595, −7.17466028897575146163433776448, −6.74534472074773064211733407631, −6.44149624567824050826581296683, −6.19818460520284543013384022758, −5.66992594967210592452856750986, −5.54722890019173235553891761096, −4.96284150914739758244404038592, −4.67532184049123285660327163647, −3.87537894885409196915233245817, −3.42307819014211425049537325394, −3.22012070837500961362961836022, −2.73441290422602185310196572047, −2.18253588002634172535927680470, −1.90459815303010201662473188351, −1.17217794914892801672586287374, −1.05219458621003520936746611663, 1.05219458621003520936746611663, 1.17217794914892801672586287374, 1.90459815303010201662473188351, 2.18253588002634172535927680470, 2.73441290422602185310196572047, 3.22012070837500961362961836022, 3.42307819014211425049537325394, 3.87537894885409196915233245817, 4.67532184049123285660327163647, 4.96284150914739758244404038592, 5.54722890019173235553891761096, 5.66992594967210592452856750986, 6.19818460520284543013384022758, 6.44149624567824050826581296683, 6.74534472074773064211733407631, 7.17466028897575146163433776448, 7.67137938126613580363559479595, 8.042568480195725916233526167866, 8.500689446833055001099449387147, 8.721218405147397565040916811109

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