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Properties

Label 2-370-5.4-c1-0-8
Degree $2$
Conductor $370$
Sign $0.929 - 0.369i$
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  + i·2-s − 1.53i·3-s − 4-s + (2.07 − 0.826i)5-s + 1.53·6-s + 4.67i·7-s − i·8-s + 0.653·9-s + (0.826 + 2.07i)10-s + 0.0451·11-s + 1.53i·12-s − 5.26i·13-s − 4.67·14-s + (−1.26 − 3.18i)15-s + 16-s + 3.60i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.884i·3-s − 0.5·4-s + (0.929 − 0.369i)5-s + 0.625·6-s + 1.76i·7-s − 0.353i·8-s + 0.217·9-s + (0.261 + 0.656i)10-s + 0.0136·11-s + 0.442i·12-s − 1.46i·13-s − 1.24·14-s + (−0.327 − 0.821i)15-s + 0.250·16-s + 0.875i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.929 - 0.369i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.929 - 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53177 + 0.293623i\)
\(L(\frac12)\) \(\approx\) \(1.53177 + 0.293623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-2.07 + 0.826i)T \)
37 \( 1 - iT \)
good3 \( 1 + 1.53iT - 3T^{2} \)
7 \( 1 - 4.67iT - 7T^{2} \)
11 \( 1 - 0.0451T + 11T^{2} \)
13 \( 1 + 5.26iT - 13T^{2} \)
17 \( 1 - 3.60iT - 17T^{2} \)
19 \( 1 - 6.22T + 19T^{2} \)
23 \( 1 - 2.20iT - 23T^{2} \)
29 \( 1 - 4.20T + 29T^{2} \)
31 \( 1 + 3.01T + 31T^{2} \)
41 \( 1 + 7.38T + 41T^{2} \)
43 \( 1 - 5.54iT - 43T^{2} \)
47 \( 1 + 4.28iT - 47T^{2} \)
53 \( 1 + 6.10iT - 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 + 4.27T + 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + 7.03iT - 73T^{2} \)
79 \( 1 - 8.72T + 79T^{2} \)
83 \( 1 - 0.880iT - 83T^{2} \)
89 \( 1 + 9.97T + 89T^{2} \)
97 \( 1 - 0.240iT - 97T^{2} \)
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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

−11.85686246804824023242198497890, −10.25123520569866193073590084424, −9.427669910625787889358484615095, −8.491750668919209217585482289195, −7.78694544251571181228283645023, −6.50316616030069130081753198319, −5.72516662209542242384205222092, −5.12598411223450584152980249523, −2.94838572005991507048950569191, −1.53160559767277343429938742611, 1.43742774170773838884226522745, 3.20060964081897222863812785550, 4.25157443192212301060702056408, 5.02875697889845731173181147272, 6.68008876983599134574278213457, 7.44305849162270656724080138365, 9.214964545292429324157589255274, 9.650743999941415576751711432666, 10.45978832234641292022842658468, 10.96974159034809080562054575032

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