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Properties

Label 2-370-37.34-c1-0-12
Degree $2$
Conductor $370$
Sign $-0.514 + 0.857i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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 + (2.38 − 1.99i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s − 3.10·6-s + (−0.187 − 0.0684i)7-s + (0.500 − 0.866i)8-s + (1.15 − 6.56i)9-s + (0.5 + 0.866i)10-s + (−1.84 + 3.18i)11-s + (2.38 + 1.99i)12-s + (−0.982 − 5.57i)13-s + (0.100 + 0.173i)14-s + (−2.92 + 1.06i)15-s + (−0.939 + 0.342i)16-s + (1.13 − 6.44i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (1.37 − 1.15i)3-s + (0.0868 + 0.492i)4-s + (−0.420 − 0.152i)5-s − 1.26·6-s + (−0.0710 − 0.0258i)7-s + (0.176 − 0.306i)8-s + (0.385 − 2.18i)9-s + (0.158 + 0.273i)10-s + (−0.554 + 0.961i)11-s + (0.687 + 0.576i)12-s + (−0.272 − 1.54i)13-s + (0.0267 + 0.0462i)14-s + (−0.754 + 0.274i)15-s + (−0.234 + 0.0855i)16-s + (0.275 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.514 + 0.857i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.514 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699310 - 1.23560i\)
\(L(\frac12)\) \(\approx\) \(0.699310 - 1.23560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-4.31 - 4.28i)T \)
good3 \( 1 + (-2.38 + 1.99i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (0.187 + 0.0684i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.84 - 3.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.982 + 5.57i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.13 + 6.44i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (-2.02 + 1.70i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-3.60 - 6.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.38 - 4.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.12T + 31T^{2} \)
41 \( 1 + (-1.67 - 9.51i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 6.97T + 43T^{2} \)
47 \( 1 + (-1.04 - 1.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.42 - 3.43i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-11.1 + 4.05i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.219 + 1.24i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.61 + 0.950i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.503 - 0.422i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 11.4T + 73T^{2} \)
79 \( 1 + (3.89 + 1.41i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.79 - 10.2i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (1.50 - 0.547i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-3.37 - 5.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.20349313261203661913258737017, −9.747512339867616086508274785999, −9.324144946938154994773199490075, −8.037977898989623416273893090178, −7.65237374096916065858869140772, −6.96309226060425723762467318581, −5.06697968427197166487034880357, −3.25807865565037454644922371554, −2.64883820595673831343729612306, −1.05633642386113766312505442807, 2.32018715363237886540124875610, 3.65103171292880976882164699122, 4.52504205577689986151700508687, 5.97333843642937106552850675713, 7.37626815509556178174503116341, 8.287711301467054845852866380381, 8.824224256072811149838799716728, 9.668078568491940489145585811951, 10.54789426954440873302101663919, 11.22473542338293355251039358851

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