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Properties

Label 2-74-37.6-c6-0-2
Degree $2$
Conductor $74$
Sign $-0.756 - 0.654i$
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 + 23.0i·3-s − 32i·4-s + (−19.9 − 19.9i)5-s + (−92.2 − 92.2i)6-s + 259.·7-s + (128 + 128i)8-s + 197.·9-s + 159.·10-s + 1.20e3i·11-s + 737.·12-s + (1.35e3 + 1.35e3i)13-s + (−1.03e3 + 1.03e3i)14-s + (459. − 459. i)15-s − 1.02e3·16-s + (−535. − 535. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + 0.853i·3-s − 0.5i·4-s + (−0.159 − 0.159i)5-s + (−0.426 − 0.426i)6-s + 0.756·7-s + (0.250 + 0.250i)8-s + 0.270·9-s + 0.159·10-s + 0.909i·11-s + 0.426·12-s + (0.617 + 0.617i)13-s + (−0.378 + 0.378i)14-s + (0.136 − 0.136i)15-s − 0.250·16-s + (−0.108 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.756 - 0.654i$
Analytic conductor: \(17.0240\)
Root analytic conductor: \(4.12601\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3),\ -0.756 - 0.654i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.476612 + 1.27934i\)
\(L(\frac12)\) \(\approx\) \(0.476612 + 1.27934i\)
\(L(4)\) 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 \( 1 + (-4.09e4 - 2.98e4i)T \)
good3 \( 1 - 23.0iT - 729T^{2} \)
5 \( 1 + (19.9 + 19.9i)T + 1.56e4iT^{2} \)
7 \( 1 - 259.T + 1.17e5T^{2} \)
11 \( 1 - 1.20e3iT - 1.77e6T^{2} \)
13 \( 1 + (-1.35e3 - 1.35e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (535. + 535. i)T + 2.41e7iT^{2} \)
19 \( 1 + (-3.31e3 - 3.31e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (8.56e3 + 8.56e3i)T + 1.48e8iT^{2} \)
29 \( 1 + (2.07e4 - 2.07e4i)T - 5.94e8iT^{2} \)
31 \( 1 + (3.98e4 - 3.98e4i)T - 8.87e8iT^{2} \)
41 \( 1 - 1.09e4iT - 4.75e9T^{2} \)
43 \( 1 + (-3.69e4 - 3.69e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 1.05e5T + 1.07e10T^{2} \)
53 \( 1 - 1.26e5T + 2.21e10T^{2} \)
59 \( 1 + (-5.49e4 - 5.49e4i)T + 4.21e10iT^{2} \)
61 \( 1 + (-1.51e5 + 1.51e5i)T - 5.15e10iT^{2} \)
67 \( 1 - 2.46e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.17e5T + 1.28e11T^{2} \)
73 \( 1 + 1.31e5iT - 1.51e11T^{2} \)
79 \( 1 + (-5.82e5 - 5.82e5i)T + 2.43e11iT^{2} \)
83 \( 1 + 3.90e5T + 3.26e11T^{2} \)
89 \( 1 + (1.29e5 - 1.29e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-1.93e5 - 1.93e5i)T + 8.32e11iT^{2} \)
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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

−14.25216518464286801396754656796, −12.62839354741861335436147593788, −11.27082053806547080182735442042, −10.23790439589363273165762659806, −9.265815048073993462482249166584, −8.116171392211318375524570752499, −6.84223731770927165048279935901, −5.13782171195047613807825479150, −4.09165958257700090282164486070, −1.59831603189152984908804243557, 0.65905257641715029333395113947, 1.95510520467528923519658705124, 3.72747749320778805583952222375, 5.76939535462625706581074945272, 7.40538446387059599329492520163, 8.120924905184945259474117668499, 9.489434644778764122115524439019, 11.00745180542259465461993151812, 11.58317601940384278169907047171, 12.97705443809707158270488249721

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