Properties

Label 2-74-37.31-c6-0-9
Degree $2$
Conductor $74$
Sign $-0.614 - 0.788i$
Analytic cond. $17.0240$
Root an. cond. $4.12601$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 4i)2-s − 0.398i·3-s + 32i·4-s + (44.3 − 44.3i)5-s + (1.59 − 1.59i)6-s − 329.·7-s + (−128 + 128i)8-s + 728.·9-s + 354.·10-s + 2.53e3i·11-s + 12.7·12-s + (−1.64e3 + 1.64e3i)13-s + (−1.31e3 − 1.31e3i)14-s + (−17.6 − 17.6i)15-s − 1.02e3·16-s + (4.01e3 − 4.01e3i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s − 0.0147i·3-s + 0.5i·4-s + (0.354 − 0.354i)5-s + (0.00738 − 0.00738i)6-s − 0.960·7-s + (−0.250 + 0.250i)8-s + 0.999·9-s + 0.354·10-s + 1.90i·11-s + 0.00738·12-s + (−0.749 + 0.749i)13-s + (−0.480 − 0.480i)14-s + (−0.00523 − 0.00523i)15-s − 0.250·16-s + (0.817 − 0.817i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.614 - 0.788i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ -0.614 - 0.788i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.829255 + 1.69828i\)
\(L(\frac12)\) \(\approx\) \(0.829255 + 1.69828i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 4i)T \)
37 \( 1 + (2.77e4 + 4.23e4i)T \)
good3 \( 1 + 0.398iT - 729T^{2} \)
5 \( 1 + (-44.3 + 44.3i)T - 1.56e4iT^{2} \)
7 \( 1 + 329.T + 1.17e5T^{2} \)
11 \( 1 - 2.53e3iT - 1.77e6T^{2} \)
13 \( 1 + (1.64e3 - 1.64e3i)T - 4.82e6iT^{2} \)
17 \( 1 + (-4.01e3 + 4.01e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (4.52e3 - 4.52e3i)T - 4.70e7iT^{2} \)
23 \( 1 + (1.01e4 - 1.01e4i)T - 1.48e8iT^{2} \)
29 \( 1 + (5.12e3 + 5.12e3i)T + 5.94e8iT^{2} \)
31 \( 1 + (-1.58e4 - 1.58e4i)T + 8.87e8iT^{2} \)
41 \( 1 + 2.25e4iT - 4.75e9T^{2} \)
43 \( 1 + (8.89e4 - 8.89e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 1.75e5T + 1.07e10T^{2} \)
53 \( 1 - 2.10e5T + 2.21e10T^{2} \)
59 \( 1 + (-1.81e5 + 1.81e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-2.99e4 - 2.99e4i)T + 5.15e10iT^{2} \)
67 \( 1 + 5.38e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.27e5T + 1.28e11T^{2} \)
73 \( 1 - 2.92e5iT - 1.51e11T^{2} \)
79 \( 1 + (-2.96e5 + 2.96e5i)T - 2.43e11iT^{2} \)
83 \( 1 + 9.73e5T + 3.26e11T^{2} \)
89 \( 1 + (-6.30e5 - 6.30e5i)T + 4.96e11iT^{2} \)
97 \( 1 + (5.24e5 - 5.24e5i)T - 8.32e11iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.69603712517246226030282894656, −12.59955574932586005833002757225, −12.16268070329603875384923626861, −9.941222644701121547967637695318, −9.512288316439003077768240016207, −7.50641818261525594982802617512, −6.75851170301951638440045419106, −5.16599089233877207843460907254, −3.97433045779745866046108506557, −1.95733330728237873588503750478, 0.60099164986997014191544515719, 2.64036971669343740672631818299, 3.86217511801144016252953134801, 5.68227034272981272374918224231, 6.67422175759376364480886422741, 8.445696717010369513973382450805, 10.06011951551581367043697283907, 10.50177484389709107519944611801, 12.05619888050228647881591698323, 13.02300219388715915312394009943

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