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Properties

Label 2-74-37.12-c1-0-0
Degree $2$
Conductor $74$
Sign $0.880 + 0.473i$
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  + (−0.766 + 0.642i)2-s + (−1.43 − 1.20i)3-s + (0.173 − 0.984i)4-s + (3.31 − 1.20i)5-s + 1.87·6-s + (0.826 − 0.300i)7-s + (0.500 + 0.866i)8-s + (0.0923 + 0.524i)9-s + (−1.76 + 3.05i)10-s + (−1.67 − 2.89i)11-s + (−1.43 + 1.20i)12-s + (−1.11 + 6.31i)13-s + (−0.439 + 0.761i)14-s + (−6.23 − 2.27i)15-s + (−0.939 − 0.342i)16-s + (−0.520 − 2.95i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.831 − 0.697i)3-s + (0.0868 − 0.492i)4-s + (1.48 − 0.540i)5-s + 0.767·6-s + (0.312 − 0.113i)7-s + (0.176 + 0.306i)8-s + (0.0307 + 0.174i)9-s + (−0.558 + 0.967i)10-s + (−0.504 − 0.874i)11-s + (−0.415 + 0.348i)12-s + (−0.308 + 1.75i)13-s + (−0.117 + 0.203i)14-s + (−1.61 − 0.586i)15-s + (−0.234 − 0.0855i)16-s + (−0.126 − 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678508 - 0.170826i\)
\(L(\frac12)\) \(\approx\) \(0.678508 - 0.170826i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (3.44 - 5.01i)T \)
good3 \( 1 + (1.43 + 1.20i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (-3.31 + 1.20i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.826 + 0.300i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (1.67 + 2.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.11 - 6.31i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.520 + 2.95i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-3.55 - 2.98i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (2.91 - 5.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.63 - 2.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.12T + 31T^{2} \)
41 \( 1 + (-1.49 + 8.45i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 5.61T + 43T^{2} \)
47 \( 1 + (-2.56 + 4.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.252 + 0.0918i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (7.15 + 2.60i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.369 + 2.09i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (8.01 - 2.91i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-10.0 - 8.45i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 8.57T + 73T^{2} \)
79 \( 1 + (-10.8 + 3.96i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-0.724 - 4.10i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (7.86 + 2.86i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-8.53 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.07589548316873670257593598046, −13.72130744303775566914454541995, −12.24059055514355753039499905888, −11.22108590087094980326940917592, −9.792830713763479052593183120255, −8.920002900746058253705129010598, −7.25307361417278530826443201261, −6.10008485594824768482219715413, −5.25838212421876824016455278450, −1.59013270866072073062704600361, 2.49377108056394432966295850798, 4.96917904069463441676932877451, 6.06633479311166729227181358383, 7.81205207965268755664204507915, 9.556268847832162792380504731987, 10.36678143118228997176234470288, 10.80560483297321943769679328476, 12.37130676594942482871804590783, 13.40741425438717766319729073544, 14.79219508229132085500120868295

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