Properties

Label 2-74-37.12-c3-0-5
Degree $2$
Conductor $74$
Sign $-0.376 + 0.926i$
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 + (−1.34 − 1.12i)3-s + (0.694 − 3.93i)4-s + (6.03 − 2.19i)5-s + 3.50·6-s + (−28.8 + 10.4i)7-s + (4.00 + 6.92i)8-s + (−4.15 − 23.5i)9-s + (−6.42 + 11.1i)10-s + (−18.0 − 31.3i)11-s + (−5.36 + 4.50i)12-s + (10.9 − 62.0i)13-s + (30.6 − 53.1i)14-s + (−10.5 − 3.84i)15-s + (−15.0 − 5.47i)16-s + (−1.92 − 10.9i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.258 − 0.216i)3-s + (0.0868 − 0.492i)4-s + (0.539 − 0.196i)5-s + 0.238·6-s + (−1.55 + 0.566i)7-s + (0.176 + 0.306i)8-s + (−0.153 − 0.873i)9-s + (−0.203 + 0.351i)10-s + (−0.495 − 0.858i)11-s + (−0.129 + 0.108i)12-s + (0.233 − 1.32i)13-s + (0.585 − 1.01i)14-s + (−0.181 − 0.0662i)15-s + (−0.234 − 0.0855i)16-s + (−0.0275 − 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.376 + 0.926i$
Analytic conductor: \(4.36614\)
Root analytic conductor: \(2.08953\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :3/2),\ -0.376 + 0.926i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.287894 - 0.427816i\)
\(L(\frac12)\) \(\approx\) \(0.287894 - 0.427816i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-198. - 106. i)T \)
good3 \( 1 + (1.34 + 1.12i)T + (4.68 + 26.5i)T^{2} \)
5 \( 1 + (-6.03 + 2.19i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (28.8 - 10.4i)T + (262. - 220. i)T^{2} \)
11 \( 1 + (18.0 + 31.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-10.9 + 62.0i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (1.92 + 10.9i)T + (-4.61e3 + 1.68e3i)T^{2} \)
19 \( 1 + (29.3 + 24.6i)T + (1.19e3 + 6.75e3i)T^{2} \)
23 \( 1 + (55.9 - 96.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (135. + 234. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 209.T + 2.97e4T^{2} \)
41 \( 1 + (77.3 - 438. i)T + (-6.47e4 - 2.35e4i)T^{2} \)
43 \( 1 - 347.T + 7.95e4T^{2} \)
47 \( 1 + (-49.0 + 84.9i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (201. + 73.2i)T + (1.14e5 + 9.56e4i)T^{2} \)
59 \( 1 + (333. + 121. i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (-124. + 706. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (384. - 139. i)T + (2.30e5 - 1.93e5i)T^{2} \)
71 \( 1 + (362. + 304. i)T + (6.21e4 + 3.52e5i)T^{2} \)
73 \( 1 - 269.T + 3.89e5T^{2} \)
79 \( 1 + (366. - 133. i)T + (3.77e5 - 3.16e5i)T^{2} \)
83 \( 1 + (85.2 + 483. i)T + (-5.37e5 + 1.95e5i)T^{2} \)
89 \( 1 + (-907. - 330. i)T + (5.40e5 + 4.53e5i)T^{2} \)
97 \( 1 + (-492. + 852. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.50162946887465853078129877319, −12.86523899086332432269914236571, −11.48945040113603953221694057586, −9.968003697351916421649955135991, −9.307317815955778801063115712231, −7.951334397161008103677998558631, −6.19836045571021410507201840055, −5.84017171178025207868071330554, −3.07745898557674619832198329644, −0.38409854742092100513090534956, 2.31541587570016459154720749121, 4.19817358654477076988889015523, 6.20033721251327843975042076552, 7.36827231611001890899346093470, 9.075518567499701141925856567015, 10.11127323631756802169069010448, 10.67737666431851724178276835424, 12.22859537394097298672743138941, 13.20412514059506075312456472533, 14.15607799947229920338728276392

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