Properties

Label 2-74-37.34-c3-0-0
Degree $2$
Conductor $74$
Sign $-0.969 + 0.243i$
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  + (1.53 + 1.28i)2-s + (−2.66 + 2.23i)3-s + (0.694 + 3.93i)4-s + (−17.8 − 6.49i)5-s − 6.96·6-s + (−11.9 − 4.34i)7-s + (−4.00 + 6.92i)8-s + (−2.58 + 14.6i)9-s + (−18.9 − 32.8i)10-s + (8.70 − 15.0i)11-s + (−10.6 − 8.94i)12-s + (1.74 + 9.89i)13-s + (−12.7 − 22.0i)14-s + (62.1 − 22.6i)15-s + (−15.0 + 5.47i)16-s + (−3.12 + 17.6i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−0.513 + 0.430i)3-s + (0.0868 + 0.492i)4-s + (−1.59 − 0.581i)5-s − 0.473·6-s + (−0.644 − 0.234i)7-s + (−0.176 + 0.306i)8-s + (−0.0957 + 0.542i)9-s + (−0.600 − 1.04i)10-s + (0.238 − 0.413i)11-s + (−0.256 − 0.215i)12-s + (0.0372 + 0.211i)13-s + (−0.242 − 0.420i)14-s + (1.06 − 0.389i)15-s + (−0.234 + 0.0855i)16-s + (−0.0445 + 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.969 + 0.243i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.969 + 0.243i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0455375 - 0.367653i\)
\(L(\frac12)\) \(\approx\) \(0.0455375 - 0.367653i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.53 - 1.28i)T \)
37 \( 1 + (-78.2 - 211. i)T \)
good3 \( 1 + (2.66 - 2.23i)T + (4.68 - 26.5i)T^{2} \)
5 \( 1 + (17.8 + 6.49i)T + (95.7 + 80.3i)T^{2} \)
7 \( 1 + (11.9 + 4.34i)T + (262. + 220. i)T^{2} \)
11 \( 1 + (-8.70 + 15.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-1.74 - 9.89i)T + (-2.06e3 + 751. i)T^{2} \)
17 \( 1 + (3.12 - 17.6i)T + (-4.61e3 - 1.68e3i)T^{2} \)
19 \( 1 + (113. - 95.2i)T + (1.19e3 - 6.75e3i)T^{2} \)
23 \( 1 + (-19.8 - 34.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-38.7 + 67.1i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 23.0T + 2.97e4T^{2} \)
41 \( 1 + (55.2 + 313. i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + 232.T + 7.95e4T^{2} \)
47 \( 1 + (-210. - 364. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-398. + 144. i)T + (1.14e5 - 9.56e4i)T^{2} \)
59 \( 1 + (610. - 222. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (52.4 + 297. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (754. + 274. i)T + (2.30e5 + 1.93e5i)T^{2} \)
71 \( 1 + (234. - 196. i)T + (6.21e4 - 3.52e5i)T^{2} \)
73 \( 1 + 994.T + 3.89e5T^{2} \)
79 \( 1 + (-698. - 254. i)T + (3.77e5 + 3.16e5i)T^{2} \)
83 \( 1 + (75.2 - 426. i)T + (-5.37e5 - 1.95e5i)T^{2} \)
89 \( 1 + (-1.11e3 + 404. i)T + (5.40e5 - 4.53e5i)T^{2} \)
97 \( 1 + (-912. - 1.58e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.93892423426352536611604324030, −13.51362888736482264222359994744, −12.40868371365234448013454119419, −11.58013488387755907067533736103, −10.47292962590739090777706332983, −8.635568881004944406157403303304, −7.68430020911779784490216200021, −6.18256214682297151006053578739, −4.64097115201612665502590044119, −3.70986688170399526833437335038, 0.20735391891843347159433154263, 3.07242638802697097684359966706, 4.40542648573788829110322144185, 6.35097953753471075573727985582, 7.21378383016436064301779403792, 8.908573879231882028807321920776, 10.58684545673197465770877654452, 11.55197839892200399994310426595, 12.24691620192246585569717845113, 13.10576227926829664163709814206

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