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Properties

Label 2-370-185.184-c1-0-17
Degree $2$
Conductor $370$
Sign $-0.0485 + 0.998i$
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  + 2-s − 3.40i·3-s + 4-s + (1.28 + 1.83i)5-s − 3.40i·6-s − 2.06i·7-s + 8-s − 8.58·9-s + (1.28 + 1.83i)10-s + 3.77·11-s − 3.40i·12-s + 2.88·13-s − 2.06i·14-s + (6.23 − 4.36i)15-s + 16-s − 5.80·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.96i·3-s + 0.5·4-s + (0.573 + 0.819i)5-s − 1.38i·6-s − 0.779i·7-s + 0.353·8-s − 2.86·9-s + (0.405 + 0.579i)10-s + 1.13·11-s − 0.982i·12-s + 0.799·13-s − 0.551i·14-s + (1.60 − 1.12i)15-s + 0.250·16-s − 1.40·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0485 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0485 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.0485 + 0.998i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.0485 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48952 - 1.56365i\)
\(L(\frac12)\) \(\approx\) \(1.48952 - 1.56365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (-1.28 - 1.83i)T \)
37 \( 1 + (-4.80 - 3.72i)T \)
good3 \( 1 + 3.40iT - 3T^{2} \)
7 \( 1 + 2.06iT - 7T^{2} \)
11 \( 1 - 3.77T + 11T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
17 \( 1 + 5.80T + 17T^{2} \)
19 \( 1 - 0.157iT - 19T^{2} \)
23 \( 1 + 5.41T + 23T^{2} \)
29 \( 1 - 4.29iT - 29T^{2} \)
31 \( 1 + 0.425iT - 31T^{2} \)
41 \( 1 - 0.923T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 0.676iT - 47T^{2} \)
53 \( 1 + 9.87iT - 53T^{2} \)
59 \( 1 - 8.47iT - 59T^{2} \)
61 \( 1 + 1.23iT - 61T^{2} \)
67 \( 1 + 6.45iT - 67T^{2} \)
71 \( 1 + 3.28T + 71T^{2} \)
73 \( 1 - 0.980iT - 73T^{2} \)
79 \( 1 + 8.04iT - 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 - 7.65iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−11.35184783741575239770738636436, −10.75831657531488888754355709432, −9.135105624412960113319579335300, −7.979862017414054766759494578752, −6.94895235210604881102903591510, −6.54763927940053896543789579421, −5.81063244608675935986800453142, −3.87859067035351649990381746799, −2.51013778297215904582479656334, −1.40785865743634044270907486701, 2.45348188311889124321522534239, 4.03531936196426417872289382423, 4.43259981050831167238375387271, 5.72881392711964161739494265295, 6.11562431643264340614257543653, 8.422259398764974047163107905114, 9.090123877560904822866958140507, 9.666974568996047762055726795705, 10.81528343649301337968421816827, 11.51206057177961330035249218580

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