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Properties

Label 2-74-37.31-c4-0-4
Degree $2$
Conductor $74$
Sign $-0.976 - 0.214i$
Analytic cond. $7.64937$
Root an. cond. $2.76575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 2i)2-s + 14.0i·3-s + 8i·4-s + (−12.0 + 12.0i)5-s + (−28.1 + 28.1i)6-s + 53.8·7-s + (−16 + 16i)8-s − 116.·9-s − 48.0·10-s − 79.3i·11-s − 112.·12-s + (−82.8 + 82.8i)13-s + (107. + 107. i)14-s + (−168. − 168. i)15-s − 64·16-s + (77.9 − 77.9i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 1.56i·3-s + 0.5i·4-s + (−0.480 + 0.480i)5-s + (−0.780 + 0.780i)6-s + 1.09·7-s + (−0.250 + 0.250i)8-s − 1.43·9-s − 0.480·10-s − 0.655i·11-s − 0.780·12-s + (−0.490 + 0.490i)13-s + (0.549 + 0.549i)14-s + (−0.749 − 0.749i)15-s − 0.250·16-s + (0.269 − 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.976 - 0.214i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ -0.976 - 0.214i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.210524 + 1.94052i\)
\(L(\frac12)\) \(\approx\) \(0.210524 + 1.94052i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 2i)T \)
37 \( 1 + (-1.30e3 - 402. i)T \)
good3 \( 1 - 14.0iT - 81T^{2} \)
5 \( 1 + (12.0 - 12.0i)T - 625iT^{2} \)
7 \( 1 - 53.8T + 2.40e3T^{2} \)
11 \( 1 + 79.3iT - 1.46e4T^{2} \)
13 \( 1 + (82.8 - 82.8i)T - 2.85e4iT^{2} \)
17 \( 1 + (-77.9 + 77.9i)T - 8.35e4iT^{2} \)
19 \( 1 + (-332. + 332. i)T - 1.30e5iT^{2} \)
23 \( 1 + (517. - 517. i)T - 2.79e5iT^{2} \)
29 \( 1 + (-158. - 158. i)T + 7.07e5iT^{2} \)
31 \( 1 + (-893. - 893. i)T + 9.23e5iT^{2} \)
41 \( 1 + 1.43e3iT - 2.82e6T^{2} \)
43 \( 1 + (308. - 308. i)T - 3.41e6iT^{2} \)
47 \( 1 + 466.T + 4.87e6T^{2} \)
53 \( 1 - 5.32e3T + 7.89e6T^{2} \)
59 \( 1 + (-2.27e3 + 2.27e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (3.57e3 + 3.57e3i)T + 1.38e7iT^{2} \)
67 \( 1 - 1.42e3iT - 2.01e7T^{2} \)
71 \( 1 + 5.35e3T + 2.54e7T^{2} \)
73 \( 1 - 9.83e3iT - 2.83e7T^{2} \)
79 \( 1 + (-5.90e3 + 5.90e3i)T - 3.89e7iT^{2} \)
83 \( 1 + 8.42e3T + 4.74e7T^{2} \)
89 \( 1 + (653. + 653. i)T + 6.27e7iT^{2} \)
97 \( 1 + (-6.63e3 + 6.63e3i)T - 8.85e7iT^{2} \)
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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

−14.51749395488043798236328645608, −13.80905564649065150672743052751, −11.71754229786042575079788439596, −11.21173467810887653527614912295, −9.868900873516496230948568114806, −8.612084421947093020375410142171, −7.33844100786484218166194991528, −5.44611196257160571628944992692, −4.49247536120116136042068517726, −3.24461337960995060654687334066, 0.930616653185245521675872236173, 2.26134698992156073196147102264, 4.49674532142581685312309718220, 5.98287124794512315967247240529, 7.58667386611426020548099493912, 8.228665430558073528018424133232, 10.16765343140611627127589864283, 11.85792321732934677936507790262, 12.06040972696429397518508131266, 13.12409202018021129198593920460

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