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Properties

Label 2-74-37.10-c3-0-0
Degree $2$
Conductor $74$
Sign $-0.847 - 0.530i$
Analytic cond. $4.36614$
Root an. cond. $2.08953$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−5.07 − 8.78i)3-s + (−1.99 − 3.46i)4-s + (3.82 + 6.62i)5-s + 20.2·6-s + (−2.78 − 4.82i)7-s + 7.99·8-s + (−37.9 + 65.7i)9-s − 15.3·10-s − 66.0·11-s + (−20.2 + 35.1i)12-s + (28.6 + 49.7i)13-s + 11.1·14-s + (38.8 − 67.2i)15-s + (−8 + 13.8i)16-s + (−0.639 + 1.10i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.975 − 1.69i)3-s + (−0.249 − 0.433i)4-s + (0.342 + 0.592i)5-s + 1.38·6-s + (−0.150 − 0.260i)7-s + 0.353·8-s + (−1.40 + 2.43i)9-s − 0.484·10-s − 1.81·11-s + (−0.487 + 0.845i)12-s + (0.612 + 1.06i)13-s + 0.212·14-s + (0.668 − 1.15i)15-s + (−0.125 + 0.216i)16-s + (−0.00912 + 0.0158i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.847 - 0.530i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.847 - 0.530i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0203035 + 0.0706452i\)
\(L(\frac12)\) \(\approx\) \(0.0203035 + 0.0706452i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
37 \( 1 + (171. + 146. i)T \)
good3 \( 1 + (5.07 + 8.78i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-3.82 - 6.62i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (2.78 + 4.82i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 66.0T + 1.33e3T^{2} \)
13 \( 1 + (-28.6 - 49.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (0.639 - 1.10i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-16.1 - 27.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 143.T + 1.21e4T^{2} \)
29 \( 1 + 102.T + 2.43e4T^{2} \)
31 \( 1 + 87.0T + 2.97e4T^{2} \)
41 \( 1 + (31.8 + 55.1i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 310.T + 7.95e4T^{2} \)
47 \( 1 + 52.7T + 1.03e5T^{2} \)
53 \( 1 + (222. - 385. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (9.36 - 16.2i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (377. + 653. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-365. - 632. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-112. - 195. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 278.T + 3.89e5T^{2} \)
79 \( 1 + (-147. - 254. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (299. - 519. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-475. + 824. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 474.T + 9.12e5T^{2} \)
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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

−14.12108951414593403898097524796, −13.51795307561698656763464723655, −12.49071283611855469421462899616, −11.18335355607883057738664512367, −10.30146000747849883113740312022, −8.243150668552579293873321468414, −7.31328049417617772924280852577, −6.38278249762317931526939542326, −5.41203087906894068017960168909, −2.01655542674722780162518115911, 0.05633372896207988072311280360, 3.29766180828592836270553868620, 4.94909333503195643492063748764, 5.71482569119551232000489407694, 8.303898729639967326049712742632, 9.503376802527719381290711123397, 10.34422448720207027467947100881, 11.02112761318288871484334100249, 12.26758670047478431239646467491, 13.32524836703865123685196673939

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