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Properties

Label 2-74-37.7-c1-0-0
Degree $2$
Conductor $74$
Sign $0.928 - 0.370i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)2-s + (1.72 + 0.627i)3-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)5-s − 1.83·6-s + (0.598 + 3.39i)7-s + (−0.500 + 0.866i)8-s + (0.278 + 0.233i)9-s + (0.939 + 1.62i)10-s + (1.40 − 2.43i)11-s + (1.72 − 0.627i)12-s + (−2.65 + 2.23i)13-s + (−1.72 − 2.98i)14-s + (0.598 − 3.39i)15-s + (0.173 − 0.984i)16-s + (−2.37 − 1.99i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.994 + 0.362i)3-s + (0.383 − 0.321i)4-s + (−0.145 − 0.827i)5-s − 0.748·6-s + (0.226 + 1.28i)7-s + (−0.176 + 0.306i)8-s + (0.0927 + 0.0777i)9-s + (0.297 + 0.514i)10-s + (0.423 − 0.733i)11-s + (0.497 − 0.181i)12-s + (−0.737 + 0.618i)13-s + (−0.460 − 0.797i)14-s + (0.154 − 0.876i)15-s + (0.0434 − 0.246i)16-s + (−0.575 − 0.483i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.928 - 0.370i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.928 - 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870674 + 0.167383i\)
\(L(\frac12)\) \(\approx\) \(0.870674 + 0.167383i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (2.56 - 5.51i)T \)
good3 \( 1 + (-1.72 - 0.627i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (0.326 + 1.85i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.598 - 3.39i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-1.40 + 2.43i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.65 - 2.23i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (2.37 + 1.99i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (6.99 + 2.54i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-0.321 - 0.557i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.08 - 1.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.90T + 31T^{2} \)
41 \( 1 + (8.13 - 6.82i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 8.30T + 43T^{2} \)
47 \( 1 + (-3.92 - 6.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.839 + 4.76i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-0.0961 + 0.545i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-5.37 + 4.50i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.0366 - 0.207i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (4.75 + 1.72i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + (0.330 + 1.87i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.21 - 4.37i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (1.68 - 9.54i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (8.52 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.93800715539082527281357025172, −13.83495500100852360719255545386, −12.35536288540429063027082934257, −11.35739359984355807894208009172, −9.610186624143061983498738984193, −8.724649244667358222488303155745, −8.417437348373399069943889553352, −6.43193263559365270436746016883, −4.71995734231250318682238379607, −2.54006356054050893070693495172, 2.31994716054009641416243844545, 3.98282362841921112276334608742, 6.81210004841551863407061843341, 7.61627413552245467125509416649, 8.659827450074097166176740484543, 10.19770910535915263496537469185, 10.79591377197980021092610537142, 12.37259948542458278860797179981, 13.56049440175055657754565301191, 14.57163821606939409590157019002

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