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Properties

Label 2-74-37.10-c7-0-8
Degree $2$
Conductor $74$
Sign $0.519 + 0.854i$
Analytic cond. $23.1164$
Root an. cond. $4.80796$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 6.92i)2-s + (−27.0 − 46.8i)3-s + (−31.9 − 55.4i)4-s + (61.0 + 105. i)5-s − 433.·6-s + (612. + 1.06e3i)7-s − 511.·8-s + (−371. + 643. i)9-s + 976.·10-s + 8.41e3·11-s + (−1.73e3 + 3.00e3i)12-s + (2.86e3 + 4.95e3i)13-s + 9.79e3·14-s + (3.30e3 − 5.72e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (−3.81e3 + 6.60e3i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.578 − 1.00i)3-s + (−0.249 − 0.433i)4-s + (0.218 + 0.378i)5-s − 0.818·6-s + (0.674 + 1.16i)7-s − 0.353·8-s + (−0.169 + 0.294i)9-s + 0.308·10-s + 1.90·11-s + (−0.289 + 0.501i)12-s + (0.361 + 0.625i)13-s + 0.953·14-s + (0.252 − 0.437i)15-s + (−0.125 + 0.216i)16-s + (−0.188 + 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.519 + 0.854i$
Analytic conductor: \(23.1164\)
Root analytic conductor: \(4.80796\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :7/2),\ 0.519 + 0.854i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.01426 - 1.13245i\)
\(L(\frac12)\) \(\approx\) \(2.01426 - 1.13245i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 + 6.92i)T \)
37 \( 1 + (1.19e5 + 2.83e5i)T \)
good3 \( 1 + (27.0 + 46.8i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-61.0 - 105. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-612. - 1.06e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 8.41e3T + 1.94e7T^{2} \)
13 \( 1 + (-2.86e3 - 4.95e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (3.81e3 - 6.60e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-6.66e3 - 1.15e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + 9.29e4T + 3.40e9T^{2} \)
29 \( 1 - 1.94e5T + 1.72e10T^{2} \)
31 \( 1 - 4.01e4T + 2.75e10T^{2} \)
41 \( 1 + (-4.26e5 - 7.37e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 1.00e6T + 2.71e11T^{2} \)
47 \( 1 + 6.11e5T + 5.06e11T^{2} \)
53 \( 1 + (-4.15e5 + 7.20e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.26e6 + 2.19e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.31e5 + 2.28e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-8.53e5 - 1.47e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-4.56e5 - 7.91e5i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 1.98e6T + 1.10e13T^{2} \)
79 \( 1 + (1.41e6 + 2.44e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (1.15e6 - 2.00e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (4.52e6 - 7.84e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.55e7T + 8.07e13T^{2} \)
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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

−12.52376871716852797659896373097, −11.92577903316085488389917954025, −11.29144129499507629743939362086, −9.627545802454106466562133026449, −8.413878564851907714225329539247, −6.56973967743199412380590551454, −5.95905142937234605423555197276, −4.14800057202212877224135851619, −2.16059720699662407091108010526, −1.18162310942227803836733542089, 1.00508528616452608598175543655, 3.92498371040600341897429673864, 4.59904749643555784916069684511, 5.93292669483692639491905300907, 7.29316910116129920005422174409, 8.768035809238113754456304402905, 9.979696272647546219108381878250, 11.06336580281585156361648487620, 12.11386264445556975146959587786, 13.70474880114004099936355195136

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