Properties

Label 2-74-37.34-c1-0-2
Degree $2$
Conductor $74$
Sign $0.966 - 0.256i$
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 + (0.972 − 0.816i)3-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)5-s + 1.27·6-s + (−1.82 − 0.665i)7-s + (−0.500 + 0.866i)8-s + (−0.240 + 1.36i)9-s + (−0.766 − 1.32i)10-s + (−0.220 + 0.381i)11-s + (0.972 + 0.816i)12-s + (−0.367 − 2.08i)13-s + (−0.972 − 1.68i)14-s + (−1.82 + 0.665i)15-s + (−0.939 + 0.342i)16-s + (−0.664 + 3.76i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (0.561 − 0.471i)3-s + (0.0868 + 0.492i)4-s + (−0.643 − 0.234i)5-s + 0.518·6-s + (−0.691 − 0.251i)7-s + (−0.176 + 0.306i)8-s + (−0.0802 + 0.455i)9-s + (−0.242 − 0.419i)10-s + (−0.0664 + 0.115i)11-s + (0.280 + 0.235i)12-s + (−0.101 − 0.578i)13-s + (−0.260 − 0.450i)14-s + (−0.472 + 0.171i)15-s + (−0.234 + 0.0855i)16-s + (−0.161 + 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19885 + 0.156430i\)
\(L(\frac12)\) \(\approx\) \(1.19885 + 0.156430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 5.67i)T \)
good3 \( 1 + (-0.972 + 0.816i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (1.43 + 0.524i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.82 + 0.665i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.220 - 0.381i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.367 + 2.08i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.664 - 3.76i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (-4.55 + 3.82i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (2.85 + 4.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.36 - 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.87T + 31T^{2} \)
41 \( 1 + (-1.80 - 10.2i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 3.84T + 43T^{2} \)
47 \( 1 + (3.84 + 6.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.62 - 0.591i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.93 - 0.702i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.89 + 10.7i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.57 - 0.572i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (7.70 - 6.46i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + 9.88T + 73T^{2} \)
79 \( 1 + (12.5 + 4.56i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.617 + 3.50i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-11.1 + 4.05i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.296 - 0.513i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.56068789180220549373427425034, −13.45893230502338783113568116388, −12.82734882210553702249328146600, −11.65408780270337539405736842590, −10.13472310669363245309328951137, −8.454404909346440631883384600082, −7.68086500005625822024114520606, −6.39590487105428743063589574086, −4.64619723666353755724299713111, −3.00192275323802422962134276611, 3.02637810980931561368589511780, 4.10194151101519860627686266291, 5.94802992945798473694835313713, 7.52473350314672694987007075073, 9.199619352110332005793130174317, 9.940305223876038655807837984685, 11.53383375375361415502524236460, 12.16741953995365953990220538600, 13.64189587310173813630305192099, 14.41065871790534941523011481958

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