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Properties

Label 2-74-37.29-c2-0-3
Degree $2$
Conductor $74$
Sign $-0.277 + 0.960i$
Analytic cond. $2.01635$
Root an. cond. $1.41998$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (−0.514 − 0.297i)3-s + (−1.73 − i)4-s + (−0.866 − 3.23i)5-s + (−0.594 + 0.594i)6-s + (5.01 − 8.68i)7-s + (−2 + 1.99i)8-s + (−4.32 − 7.48i)9-s − 4.73·10-s + 8.99i·11-s + (0.594 + 1.02i)12-s + (3.55 + 13.2i)13-s + (−10.0 − 10.0i)14-s + (−0.514 + 1.92i)15-s + (1.99 + 3.46i)16-s + (25.4 + 6.83i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.171 − 0.0990i)3-s + (−0.433 − 0.250i)4-s + (−0.173 − 0.646i)5-s + (−0.0990 + 0.0990i)6-s + (0.715 − 1.24i)7-s + (−0.250 + 0.249i)8-s + (−0.480 − 0.832i)9-s − 0.473·10-s + 0.817i·11-s + (0.0495 + 0.0857i)12-s + (0.273 + 1.02i)13-s + (−0.715 − 0.715i)14-s + (−0.0343 + 0.128i)15-s + (0.124 + 0.216i)16-s + (1.49 + 0.401i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(2.01635\)
Root analytic conductor: \(1.41998\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1),\ -0.277 + 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.737863 - 0.981275i\)
\(L(\frac12)\) \(\approx\) \(0.737863 - 0.981275i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
37 \( 1 + (36.9 + 1.90i)T \)
good3 \( 1 + (0.514 + 0.297i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (0.866 + 3.23i)T + (-21.6 + 12.5i)T^{2} \)
7 \( 1 + (-5.01 + 8.68i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 - 8.99iT - 121T^{2} \)
13 \( 1 + (-3.55 - 13.2i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (-25.4 - 6.83i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (3.45 + 12.8i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (9.01 - 9.01i)T - 529iT^{2} \)
29 \( 1 + (-38.3 - 38.3i)T + 841iT^{2} \)
31 \( 1 + (15.4 + 15.4i)T + 961iT^{2} \)
41 \( 1 + (-12.7 - 7.34i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-19.9 + 19.9i)T - 1.84e3iT^{2} \)
47 \( 1 + 62.2T + 2.20e3T^{2} \)
53 \( 1 + (14.6 + 25.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (30.1 + 8.08i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-75.7 + 20.3i)T + (3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-51.3 - 29.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (5.50 - 9.54i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 35.9iT - 5.32e3T^{2} \)
79 \( 1 + (16.0 + 59.8i)T + (-5.40e3 + 3.12e3i)T^{2} \)
83 \( 1 + (-38.9 - 67.4i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (34.7 - 129. i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (45.5 - 45.5i)T - 9.40e3iT^{2} \)
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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

−14.03495035751881897944345428929, −12.67208255852538926742119974826, −11.89864212838734474968281410791, −10.82569231544612077125106071596, −9.634840678223358980787509379614, −8.371608384137104675515896621761, −6.88270460360166215763233674511, −5.00325734006819637408779381647, −3.81811121908059596922015516862, −1.22144517551066268921804337439, 3.01204521674564746820762768910, 5.21348674058600326819328534089, 6.00450654154963775299651374982, 7.86529440449455800070119994973, 8.488037030139252141410508177424, 10.26476608027755036524262273356, 11.41134130662632405794842634489, 12.44380688833524011416210499850, 13.97601999646359292067233898324, 14.61438061389226712632913399726

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