Properties

Label 2-74-37.29-c4-0-13
Degree $2$
Conductor $74$
Sign $-0.247 - 0.968i$
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  + (0.732 − 2.73i)2-s + (−13.4 − 7.78i)3-s + (−6.92 − 4i)4-s + (−9.88 − 36.8i)5-s + (−31.1 + 31.1i)6-s + (23.7 − 41.0i)7-s + (−16 + 15.9i)8-s + (80.6 + 139. i)9-s − 108.·10-s + 147. i·11-s + (62.2 + 107. i)12-s + (−49.3 − 184. i)13-s + (−94.8 − 94.8i)14-s + (−153. + 574. i)15-s + (31.9 + 55.4i)16-s + (45.3 + 12.1i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−1.49 − 0.864i)3-s + (−0.433 − 0.250i)4-s + (−0.395 − 1.47i)5-s + (−0.864 + 0.864i)6-s + (0.483 − 0.838i)7-s + (−0.250 + 0.249i)8-s + (0.995 + 1.72i)9-s − 1.08·10-s + 1.21i·11-s + (0.432 + 0.748i)12-s + (−0.291 − 1.08i)13-s + (−0.483 − 0.483i)14-s + (−0.683 + 2.55i)15-s + (0.124 + 0.216i)16-s + (0.156 + 0.0420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.247 - 0.968i$
Analytic conductor: \(7.64937\)
Root analytic conductor: \(2.76575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :2),\ -0.247 - 0.968i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.365272 + 0.470206i\)
\(L(\frac12)\) \(\approx\) \(0.365272 + 0.470206i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.732 + 2.73i)T \)
37 \( 1 + (1.21e3 - 631. i)T \)
good3 \( 1 + (13.4 + 7.78i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (9.88 + 36.8i)T + (-541. + 312.5i)T^{2} \)
7 \( 1 + (-23.7 + 41.0i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 - 147. iT - 1.46e4T^{2} \)
13 \( 1 + (49.3 + 184. i)T + (-2.47e4 + 1.42e4i)T^{2} \)
17 \( 1 + (-45.3 - 12.1i)T + (7.23e4 + 4.17e4i)T^{2} \)
19 \( 1 + (7.72 + 28.8i)T + (-1.12e5 + 6.51e4i)T^{2} \)
23 \( 1 + (-608. + 608. i)T - 2.79e5iT^{2} \)
29 \( 1 + (320. + 320. i)T + 7.07e5iT^{2} \)
31 \( 1 + (-455. - 455. i)T + 9.23e5iT^{2} \)
41 \( 1 + (2.28e3 + 1.31e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (2.34e3 - 2.34e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 3.56e3T + 4.87e6T^{2} \)
53 \( 1 + (1.55e3 + 2.69e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (2.16e3 + 580. i)T + (1.04e7 + 6.05e6i)T^{2} \)
61 \( 1 + (-1.99e3 + 535. i)T + (1.19e7 - 6.92e6i)T^{2} \)
67 \( 1 + (843. + 486. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (610. - 1.05e3i)T + (-1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + 3.04e3iT - 2.83e7T^{2} \)
79 \( 1 + (-2.89e3 - 1.08e4i)T + (-3.37e7 + 1.94e7i)T^{2} \)
83 \( 1 + (-853. - 1.47e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (238. - 888. i)T + (-5.43e7 - 3.13e7i)T^{2} \)
97 \( 1 + (1.22e4 - 1.22e4i)T - 8.85e7iT^{2} \)
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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

−12.54032396311573657912268832480, −12.30160021742657132351676230704, −11.08334987851180911365116177032, −10.08364279577964063801982869556, −8.252278946857178388441841114037, −7.03029963368094047180829320848, −5.20508542751040722280803520835, −4.61181096918232580398217615895, −1.40235525815879813764105459176, −0.38635200457047483703927976701, 3.53747933535987249815946209212, 5.13187543698694692689048450716, 6.12838385000209443694932950450, 7.15324839012727634348893660435, 8.989828244364084379521456598306, 10.43278974435847321191995769242, 11.40674779038629146893098447308, 11.86688520559846165433063947027, 13.84982558873517784054761762860, 15.02462359393291991242425202930

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