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Properties

Label 2-74-37.31-c4-0-0
Degree $2$
Conductor $74$
Sign $-0.876 + 0.481i$
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 + 13.0i·3-s + 8i·4-s + (−23.5 + 23.5i)5-s + (26.1 − 26.1i)6-s − 19.7·7-s + (16 − 16i)8-s − 90.5·9-s + 94.1·10-s − 152. i·11-s − 104.·12-s + (144. − 144. i)13-s + (39.5 + 39.5i)14-s + (−308. − 308. i)15-s − 64·16-s + (−104. + 104. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 1.45i·3-s + 0.5i·4-s + (−0.941 + 0.941i)5-s + (0.727 − 0.727i)6-s − 0.403·7-s + (0.250 − 0.250i)8-s − 1.11·9-s + 0.941·10-s − 1.25i·11-s − 0.727·12-s + (0.853 − 0.853i)13-s + (0.201 + 0.201i)14-s + (−1.37 − 1.37i)15-s − 0.250·16-s + (−0.362 + 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.876 + 0.481i$
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: \((2,\ 74,\ (\ :2),\ -0.876 + 0.481i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0635284 - 0.247467i\)
\(L(\frac12)\) \(\approx\) \(0.0635284 - 0.247467i\)
\(L(3)\) 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 \( 1 + (1.25e3 - 558. i)T \)
good3 \( 1 - 13.0iT - 81T^{2} \)
5 \( 1 + (23.5 - 23.5i)T - 625iT^{2} \)
7 \( 1 + 19.7T + 2.40e3T^{2} \)
11 \( 1 + 152. iT - 1.46e4T^{2} \)
13 \( 1 + (-144. + 144. i)T - 2.85e4iT^{2} \)
17 \( 1 + (104. - 104. i)T - 8.35e4iT^{2} \)
19 \( 1 + (227. - 227. i)T - 1.30e5iT^{2} \)
23 \( 1 + (346. - 346. i)T - 2.79e5iT^{2} \)
29 \( 1 + (831. + 831. i)T + 7.07e5iT^{2} \)
31 \( 1 + (-237. - 237. i)T + 9.23e5iT^{2} \)
41 \( 1 - 316. iT - 2.82e6T^{2} \)
43 \( 1 + (-2.17e3 + 2.17e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 613.T + 4.87e6T^{2} \)
53 \( 1 + 2.53e3T + 7.89e6T^{2} \)
59 \( 1 + (2.73e3 - 2.73e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-1.63e3 - 1.63e3i)T + 1.38e7iT^{2} \)
67 \( 1 - 6.63e3iT - 2.01e7T^{2} \)
71 \( 1 + 9.61e3T + 2.54e7T^{2} \)
73 \( 1 + 5.98e3iT - 2.83e7T^{2} \)
79 \( 1 + (-7.62e3 + 7.62e3i)T - 3.89e7iT^{2} \)
83 \( 1 + 2.35e3T + 4.74e7T^{2} \)
89 \( 1 + (-4.76e3 - 4.76e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (-113. + 113. i)T - 8.85e7iT^{2} \)
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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

−14.80083187139975869977925944634, −13.38193186829983313233538468476, −11.72056772052007056613310926020, −10.81320949544407149509537557456, −10.34133776789056636328460518747, −8.949448344882931489023996318881, −7.88175444510900974664701523884, −6.00221715657631957029195891817, −3.88202723334510770710677957698, −3.28233409351122936408875641431, 0.15493793344326275676906982287, 1.73178414000045282133590468960, 4.50170811890298264778476989304, 6.42519270047644667183934474920, 7.29919792309983182301431228676, 8.315229210924200192662164684726, 9.312207425844128359837565861720, 11.21157662198762738840076237569, 12.38450166396094552814859636147, 12.93959476068913057099310694037

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