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Properties

Label 2-370-185.103-c1-0-14
Degree $2$
Conductor $370$
Sign $0.295 + 0.955i$
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  + (0.5 + 0.866i)2-s + (−0.418 − 1.56i)3-s + (−0.499 + 0.866i)4-s + (−2.09 + 0.785i)5-s + (1.14 − 1.14i)6-s + (−0.298 − 1.11i)7-s − 0.999·8-s + (0.332 − 0.192i)9-s + (−1.72 − 1.42i)10-s − 5.41i·11-s + (1.56 + 0.418i)12-s + (2.03 − 3.53i)13-s + (0.816 − 0.816i)14-s + (2.10 + 2.94i)15-s + (−0.5 − 0.866i)16-s + (3.37 − 1.94i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.241 − 0.901i)3-s + (−0.249 + 0.433i)4-s + (−0.936 + 0.351i)5-s + (0.466 − 0.466i)6-s + (−0.112 − 0.421i)7-s − 0.353·8-s + (0.110 − 0.0640i)9-s + (−0.546 − 0.449i)10-s − 1.63i·11-s + (0.450 + 0.120i)12-s + (0.565 − 0.979i)13-s + (0.218 − 0.218i)14-s + (0.543 + 0.759i)15-s + (−0.125 − 0.216i)16-s + (0.818 − 0.472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.295 + 0.955i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855064 - 0.630272i\)
\(L(\frac12)\) \(\approx\) \(0.855064 - 0.630272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (2.09 - 0.785i)T \)
37 \( 1 + (-5.44 + 2.71i)T \)
good3 \( 1 + (0.418 + 1.56i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.298 + 1.11i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 5.41iT - 11T^{2} \)
13 \( 1 + (-2.03 + 3.53i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.37 + 1.94i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.80 - 1.55i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.383T + 23T^{2} \)
29 \( 1 + (5.82 - 5.82i)T - 29iT^{2} \)
31 \( 1 + (3.43 + 3.43i)T + 31iT^{2} \)
41 \( 1 + (-7.78 - 4.49i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.57T + 43T^{2} \)
47 \( 1 + (3.06 - 3.06i)T - 47iT^{2} \)
53 \( 1 + (2.04 - 7.61i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.84 - 10.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (8.58 - 2.30i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-10.0 + 2.68i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.42 + 4.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.61 + 9.61i)T - 73iT^{2} \)
79 \( 1 + (-8.53 + 2.28i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-3.22 + 12.0i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (14.3 + 3.83i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 4.38iT - 97T^{2} \)
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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

−11.23100075907357347849303636615, −10.68019807764705080146047244173, −9.038143153201382555794292699632, −7.898402685305785076869102758647, −7.57303160758478095273882811250, −6.37992827848594888241428534766, −5.74011819641176422436688879509, −4.06183228798286887208389991787, −3.16537692305447661029143860570, −0.68550071801466159707322106144, 1.98918680758171453997992755989, 3.88719821021626974082614010069, 4.32341872583689785719467369530, 5.31810772011321225624529761869, 6.75841446513682052795287998982, 7.969601759892271259416501352623, 9.194801892901113872604896378800, 9.777478402633465820185895608831, 10.86391292686989293654737963345, 11.43836133756113644667001785233

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