Properties

Label 2-74-37.34-c5-0-16
Degree $2$
Conductor $74$
Sign $-0.328 + 0.944i$
Analytic cond. $11.8684$
Root an. cond. $3.44505$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.06 + 2.57i)2-s + (21.8 − 18.3i)3-s + (2.77 + 15.7i)4-s + (−84.9 − 30.9i)5-s + 114.·6-s + (−181. − 65.9i)7-s + (−32.0 + 55.4i)8-s + (98.9 − 561. i)9-s + (−180. − 313. i)10-s + (234. − 406. i)11-s + (349. + 293. i)12-s + (69.1 + 392. i)13-s + (−385. − 667. i)14-s + (−2.42e3 + 881. i)15-s + (−240. + 87.5i)16-s + (−113. + 645. i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (1.40 − 1.17i)3-s + (0.0868 + 0.492i)4-s + (−1.51 − 0.552i)5-s + 1.29·6-s + (−1.39 − 0.508i)7-s + (−0.176 + 0.306i)8-s + (0.407 − 2.31i)9-s + (−0.571 − 0.989i)10-s + (0.584 − 1.01i)11-s + (0.700 + 0.587i)12-s + (0.113 + 0.643i)13-s + (−0.525 − 0.910i)14-s + (−2.77 + 1.01i)15-s + (−0.234 + 0.0855i)16-s + (−0.0955 + 0.541i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.328 + 0.944i$
Analytic conductor: \(11.8684\)
Root analytic conductor: \(3.44505\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :5/2),\ -0.328 + 0.944i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.27199 - 1.78829i\)
\(L(\frac12)\) \(\approx\) \(1.27199 - 1.78829i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.06 - 2.57i)T \)
37 \( 1 + (8.11e3 + 1.86e3i)T \)
good3 \( 1 + (-21.8 + 18.3i)T + (42.1 - 239. i)T^{2} \)
5 \( 1 + (84.9 + 30.9i)T + (2.39e3 + 2.00e3i)T^{2} \)
7 \( 1 + (181. + 65.9i)T + (1.28e4 + 1.08e4i)T^{2} \)
11 \( 1 + (-234. + 406. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-69.1 - 392. i)T + (-3.48e5 + 1.26e5i)T^{2} \)
17 \( 1 + (113. - 645. i)T + (-1.33e6 - 4.85e5i)T^{2} \)
19 \( 1 + (-1.50e3 + 1.26e3i)T + (4.29e5 - 2.43e6i)T^{2} \)
23 \( 1 + (-1.02e3 - 1.77e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-2.76e3 + 4.79e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 5.05e3T + 2.86e7T^{2} \)
41 \( 1 + (2.91e3 + 1.65e4i)T + (-1.08e8 + 3.96e7i)T^{2} \)
43 \( 1 + 3.01e3T + 1.47e8T^{2} \)
47 \( 1 + (-2.80e3 - 4.86e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (2.10e4 - 7.64e3i)T + (3.20e8 - 2.68e8i)T^{2} \)
59 \( 1 + (-2.15e4 + 7.86e3i)T + (5.47e8 - 4.59e8i)T^{2} \)
61 \( 1 + (-1.72e3 - 9.80e3i)T + (-7.93e8 + 2.88e8i)T^{2} \)
67 \( 1 + (-3.35e3 - 1.22e3i)T + (1.03e9 + 8.67e8i)T^{2} \)
71 \( 1 + (-5.58e4 + 4.68e4i)T + (3.13e8 - 1.77e9i)T^{2} \)
73 \( 1 + 4.35e4T + 2.07e9T^{2} \)
79 \( 1 + (1.88e4 + 6.87e3i)T + (2.35e9 + 1.97e9i)T^{2} \)
83 \( 1 + (1.78e4 - 1.01e5i)T + (-3.70e9 - 1.34e9i)T^{2} \)
89 \( 1 + (-2.94e4 + 1.07e4i)T + (4.27e9 - 3.58e9i)T^{2} \)
97 \( 1 + (1.25e4 + 2.18e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.47295737165562435165445036315, −12.47681453911398392798855231800, −11.65879334444233571635826819700, −9.222245065248785271139356749739, −8.373987480717069899624949930787, −7.33726541116505734610537087674, −6.53186562977314428217983539609, −3.89331530471554407107012083855, −3.19674452735504801201939077883, −0.69816208490494782817495397593, 2.95984724576865327104332846515, 3.45343594072923500774240447261, 4.66867687371064484222384488727, 6.98591940318661131864837502151, 8.373815599215111444326821891252, 9.617811814656202373652625109265, 10.34109336820744190402697371835, 11.80058473369193821130100032255, 12.82514632581501015869336025023, 14.23030317924683425454744026516

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