Properties

Label 2-370-185.88-c1-0-14
Degree $2$
Conductor $370$
Sign $-0.737 + 0.675i$
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 + (−0.465 − 0.124i)3-s + (−0.499 − 0.866i)4-s + (−2.03 − 0.917i)5-s + (−0.341 + 0.341i)6-s + (4.19 + 1.12i)7-s − 0.999·8-s + (−2.39 − 1.38i)9-s + (−1.81 + 1.30i)10-s − 3.56i·11-s + (0.124 + 0.465i)12-s + (−2.59 − 4.48i)13-s + (3.07 − 3.07i)14-s + (0.835 + 0.682i)15-s + (−0.5 + 0.866i)16-s + (−0.739 − 0.426i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.269 − 0.0720i)3-s + (−0.249 − 0.433i)4-s + (−0.911 − 0.410i)5-s + (−0.139 + 0.139i)6-s + (1.58 + 0.425i)7-s − 0.353·8-s + (−0.798 − 0.461i)9-s + (−0.573 + 0.413i)10-s − 1.07i·11-s + (0.0360 + 0.134i)12-s + (−0.718 − 1.24i)13-s + (0.821 − 0.821i)14-s + (0.215 + 0.176i)15-s + (−0.125 + 0.216i)16-s + (−0.179 − 0.103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.737 + 0.675i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406043 - 1.04513i\)
\(L(\frac12)\) \(\approx\) \(0.406043 - 1.04513i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.917i)T \)
37 \( 1 + (2.05 + 5.72i)T \)
good3 \( 1 + (0.465 + 0.124i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-4.19 - 1.12i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 3.56iT - 11T^{2} \)
13 \( 1 + (2.59 + 4.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.739 + 0.426i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.530 + 1.98i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 5.38T + 23T^{2} \)
29 \( 1 + (2.49 - 2.49i)T - 29iT^{2} \)
31 \( 1 + (-3.87 - 3.87i)T + 31iT^{2} \)
41 \( 1 + (-3.80 + 2.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + (-6.14 + 6.14i)T - 47iT^{2} \)
53 \( 1 + (-12.8 + 3.44i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.39 - 1.71i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.12 - 7.94i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.47 + 5.50i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.748 + 1.29i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.61 + 3.61i)T - 73iT^{2} \)
79 \( 1 + (0.864 - 3.22i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (15.3 - 4.11i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.96 - 11.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 8.46iT - 97T^{2} \)
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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

−11.15202729333172987556036745001, −10.65396522799608443832896608867, −8.979869859463356352767696674404, −8.396121994136549469852933563364, −7.49993683150140844709260885184, −5.70134927569128932495701406216, −5.17555046750981673011231786102, −3.94183787967552507313462049769, −2.63975869395839692296595985372, −0.71204308536680789555606182039, 2.25251776731401142563249293995, 4.30249887586096972616749292000, 4.56022002657637052621313086986, 5.93985791139258209613255618706, 7.29161905745355315940658291143, 7.72917278451434603347636065405, 8.618913003476963939383616355103, 10.01238427753312840385122635697, 11.15632564967706447323566247375, 11.69639150423451052316400281098

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