Properties

Label 2-74-37.10-c3-0-2
Degree $2$
Conductor $74$
Sign $0.724 - 0.689i$
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 + (−1.35 − 2.35i)3-s + (−1.99 − 3.46i)4-s + (0.388 + 0.672i)5-s + 5.43·6-s + (11.9 + 20.6i)7-s + 7.99·8-s + (9.80 − 16.9i)9-s − 1.55·10-s + 52.4·11-s + (−5.43 + 9.41i)12-s + (34.3 + 59.4i)13-s − 47.7·14-s + (1.05 − 1.82i)15-s + (−8 + 13.8i)16-s + (−44.7 + 77.4i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.261 − 0.452i)3-s + (−0.249 − 0.433i)4-s + (0.0347 + 0.0601i)5-s + 0.369·6-s + (0.645 + 1.11i)7-s + 0.353·8-s + (0.363 − 0.629i)9-s − 0.0490·10-s + 1.43·11-s + (−0.130 + 0.226i)12-s + (0.732 + 1.26i)13-s − 0.912·14-s + (0.0181 − 0.0314i)15-s + (−0.125 + 0.216i)16-s + (−0.638 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.724 - 0.689i$
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: \((2,\ 74,\ (\ :3/2),\ 0.724 - 0.689i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.18400 + 0.473189i\)
\(L(\frac12)\) \(\approx\) \(1.18400 + 0.473189i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-184. + 129. i)T \)
good3 \( 1 + (1.35 + 2.35i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-0.388 - 0.672i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-11.9 - 20.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 52.4T + 1.33e3T^{2} \)
13 \( 1 + (-34.3 - 59.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (44.7 - 77.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (47.2 + 81.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 135.T + 1.21e4T^{2} \)
29 \( 1 - 172.T + 2.43e4T^{2} \)
31 \( 1 + 286.T + 2.97e4T^{2} \)
41 \( 1 + (-80.2 - 138. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 459.T + 7.95e4T^{2} \)
47 \( 1 + 57.4T + 1.03e5T^{2} \)
53 \( 1 + (153. - 266. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (93.4 - 161. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-266. - 462. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (92.1 + 159. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (98.9 + 171. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 62.7T + 3.89e5T^{2} \)
79 \( 1 + (418. + 724. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-438. + 758. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-545. + 944. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.43e3T + 9.12e5T^{2} \)
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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

−14.59616345802997086460432484537, −13.12115808811198153447453284495, −11.94238990748640716612064465763, −11.06220408120390166397135660459, −9.116182232222408716803415691475, −8.728732061097308555022508823922, −6.81608064480606909556391887780, −6.21471716971158468806738985608, −4.38104399999551971787853362999, −1.57162865213331440461807386484, 1.23076582867424794785665752118, 3.70739109035617461649988587880, 4.95733764336215381822847515229, 7.01807297975675956518162916649, 8.313737158038610013940858385856, 9.647626442600421775086814721589, 10.79757419466006722446839627765, 11.26567048537588382966101318152, 12.83664110652213128243438984185, 13.78621141169631479147500823826

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