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Properties

Label 2-74-37.7-c1-0-1
Degree $2$
Conductor $74$
Sign $-0.141 + 0.989i$
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 + (−2.04 − 0.746i)3-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)5-s + 2.18·6-s + (−0.711 − 4.03i)7-s + (−0.500 + 0.866i)8-s + (1.34 + 1.12i)9-s + (0.939 + 1.62i)10-s + (−1.67 + 2.89i)11-s + (−2.04 + 0.746i)12-s + (1.48 − 1.24i)13-s + (2.04 + 3.55i)14-s + (−0.711 + 4.03i)15-s + (0.173 − 0.984i)16-s + (3.40 + 2.85i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−1.18 − 0.430i)3-s + (0.383 − 0.321i)4-s + (−0.145 − 0.827i)5-s + 0.890·6-s + (−0.269 − 1.52i)7-s + (−0.176 + 0.306i)8-s + (0.448 + 0.376i)9-s + (0.297 + 0.514i)10-s + (−0.503 + 0.872i)11-s + (−0.591 + 0.215i)12-s + (0.411 − 0.345i)13-s + (0.547 + 0.948i)14-s + (−0.183 + 1.04i)15-s + (0.0434 − 0.246i)16-s + (0.826 + 0.693i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.141 + 0.989i$
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: \((2,\ 74,\ (\ :1/2),\ -0.141 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274170 - 0.316212i\)
\(L(\frac12)\) \(\approx\) \(0.274170 - 0.316212i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.50 + 4.97i)T \)
good3 \( 1 + (2.04 + 0.746i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (0.326 + 1.85i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.711 + 4.03i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (1.67 - 2.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.48 + 1.24i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-3.40 - 2.85i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-0.0932 - 0.0339i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (4.76 + 8.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.387 - 0.670i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
41 \( 1 + (-4.36 + 3.66i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 1.11T + 43T^{2} \)
47 \( 1 + (-3.55 - 6.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.142 + 0.806i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-1.40 + 7.97i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (8.19 - 6.87i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.28 - 12.9i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-2.79 - 1.01i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 2.55T + 73T^{2} \)
79 \( 1 + (0.943 + 5.35i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.504 - 0.423i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-0.612 + 3.47i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.81 + 3.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.27542445170169492734526780088, −12.84326461246889491827318266382, −12.19105507375832775706603481792, −10.69192068657384834659229704462, −10.12952458195477464512602908113, −8.323445443212100885171271875574, −7.19963607668011482630723794490, −6.05282121044386137334040043395, −4.51575352084716419805689197947, −0.815260982004608777935867505137, 3.01016367370750207830118677858, 5.45694800639675010816529130081, 6.36218511913677205285461199481, 8.089896819122573072080993803306, 9.487346587602963913040368895972, 10.54940578569435887662803045323, 11.61038033644044462711472167668, 11.97877199029441675889044077185, 13.77243781217346279272698136559, 15.43580485356743638354310362955

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