Properties

Label 4-1815e2-1.1-c3e2-0-1
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $11467.9$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 6·3-s − 11·4-s − 10·5-s + 6·6-s + 4·7-s − 15·8-s + 27·9-s − 10·10-s − 66·12-s + 90·13-s + 4·14-s − 60·15-s + 61·16-s + 16·17-s + 27·18-s + 170·19-s + 110·20-s + 24·21-s − 124·23-s − 90·24-s + 75·25-s + 90·26-s + 108·27-s − 44·28-s + 158·29-s − 60·30-s + ⋯
L(s)  = 1  + 0.353·2-s + 1.15·3-s − 1.37·4-s − 0.894·5-s + 0.408·6-s + 0.215·7-s − 0.662·8-s + 9-s − 0.316·10-s − 1.58·12-s + 1.92·13-s + 0.0763·14-s − 1.03·15-s + 0.953·16-s + 0.228·17-s + 0.353·18-s + 2.05·19-s + 1.22·20-s + 0.249·21-s − 1.12·23-s − 0.765·24-s + 3/5·25-s + 0.678·26-s + 0.769·27-s − 0.296·28-s + 1.01·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(11467.9\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.836613612\)
\(L(\frac12)\) \(\approx\) \(5.836613612\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
5$C_1$ \( ( 1 + p T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 4 T + 622 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 90 T + 6402 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 16 T + 1662 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 170 T + 994 p T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 124 T + 12878 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 158 T + 50106 T^{2} - 158 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 60 T + 59870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 372 T + 82590 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 38 T + 37410 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 12 p T + 160230 T^{2} - 12 p^{4} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 224 T + 466 p T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 472 T + 190182 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 248 T + 181334 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 72 T - 107850 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 744 T + 738822 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 2060 T + 1768494 T^{2} - 2060 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 486 T + 822786 T^{2} - 486 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 642 T + 691166 T^{2} + 642 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 286 T + 392750 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 244 T + 1355190 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 168 T + 1053870 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.975919455955815981495464874013, −8.626895943751117484650205134830, −8.329416802262843097606747714239, −8.152947976751197832704830214692, −7.66648913910745085004576431271, −7.32152660429628208049124838681, −6.80865577436549044515143411398, −6.31930385897685270299856268448, −5.64792002313435453755987444478, −5.50761729835406809003913038289, −4.75949872444302853093417180656, −4.58659632820078598924933687549, −3.84712494123525724871669034581, −3.82699095849970679249022270877, −3.30519085923496854215335388401, −3.07576153527720861045827526549, −2.19846725010747700947176833085, −1.52701016907896674242472741761, −0.790390644580710781391790215407, −0.66931489168708182584882321953, 0.66931489168708182584882321953, 0.790390644580710781391790215407, 1.52701016907896674242472741761, 2.19846725010747700947176833085, 3.07576153527720861045827526549, 3.30519085923496854215335388401, 3.82699095849970679249022270877, 3.84712494123525724871669034581, 4.58659632820078598924933687549, 4.75949872444302853093417180656, 5.50761729835406809003913038289, 5.64792002313435453755987444478, 6.31930385897685270299856268448, 6.80865577436549044515143411398, 7.32152660429628208049124838681, 7.66648913910745085004576431271, 8.152947976751197832704830214692, 8.329416802262843097606747714239, 8.626895943751117484650205134830, 8.975919455955815981495464874013

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