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Properties

Label 2-74-37.6-c6-0-3
Degree $2$
Conductor $74$
Sign $-0.0264 - 0.999i$
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 − 7.34i·3-s − 32i·4-s + (166. + 166. i)5-s + (29.3 + 29.3i)6-s − 272.·7-s + (128 + 128i)8-s + 675.·9-s − 1.33e3·10-s − 847. i·11-s − 235.·12-s + (60.3 + 60.3i)13-s + (1.09e3 − 1.09e3i)14-s + (1.22e3 − 1.22e3i)15-s − 1.02e3·16-s + (296. + 296. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.272i·3-s − 0.5i·4-s + (1.33 + 1.33i)5-s + (0.136 + 0.136i)6-s − 0.795·7-s + (0.250 + 0.250i)8-s + 0.925·9-s − 1.33·10-s − 0.636i·11-s − 0.136·12-s + (0.0274 + 0.0274i)13-s + (0.397 − 0.397i)14-s + (0.362 − 0.362i)15-s − 0.250·16-s + (0.0603 + 0.0603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0264 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0264 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.0264 - 0.999i$
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: \((2,\ 74,\ (\ :3),\ -0.0264 - 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.17453 + 1.20599i\)
\(L(\frac12)\) \(\approx\) \(1.17453 + 1.20599i\)
\(L(4)\) 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 \( 1 + (-5.51e3 - 5.03e4i)T \)
good3 \( 1 + 7.34iT - 729T^{2} \)
5 \( 1 + (-166. - 166. i)T + 1.56e4iT^{2} \)
7 \( 1 + 272.T + 1.17e5T^{2} \)
11 \( 1 + 847. iT - 1.77e6T^{2} \)
13 \( 1 + (-60.3 - 60.3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-296. - 296. i)T + 2.41e7iT^{2} \)
19 \( 1 + (-6.09e3 - 6.09e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (-1.72e3 - 1.72e3i)T + 1.48e8iT^{2} \)
29 \( 1 + (2.99e4 - 2.99e4i)T - 5.94e8iT^{2} \)
31 \( 1 + (645. - 645. i)T - 8.87e8iT^{2} \)
41 \( 1 - 3.80e4iT - 4.75e9T^{2} \)
43 \( 1 + (-7.99e4 - 7.99e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 1.77e5T + 1.07e10T^{2} \)
53 \( 1 + 2.72e4T + 2.21e10T^{2} \)
59 \( 1 + (2.36e5 + 2.36e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (-1.97e5 + 1.97e5i)T - 5.15e10iT^{2} \)
67 \( 1 - 2.48e4iT - 9.04e10T^{2} \)
71 \( 1 + 2.79e5T + 1.28e11T^{2} \)
73 \( 1 - 2.86e5iT - 1.51e11T^{2} \)
79 \( 1 + (4.46e5 + 4.46e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 5.68e5T + 3.26e11T^{2} \)
89 \( 1 + (-2.15e4 + 2.15e4i)T - 4.96e11iT^{2} \)
97 \( 1 + (1.13e6 + 1.13e6i)T + 8.32e11iT^{2} \)
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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

−13.79419697446779411286545189150, −12.85392134251950382704678899307, −11.02073273764321618493428380225, −10.03868417197351088195045593909, −9.385498236304691684294319518121, −7.50285426355383497810431556074, −6.57173852743826195070516457341, −5.71967014349455135721820624598, −3.17311190652180442059303584763, −1.53147428679486385652095222233, 0.818717592000958260718787151815, 2.19299570570007984322613501060, 4.27535519001825164512811135363, 5.64643948188911322522225092341, 7.29246718248545008114332183511, 9.144772943993489082763291708079, 9.489709986599086152673339257374, 10.44248999371811197943572807787, 12.21205688003190083915272742241, 12.98260326470942683498518174724

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