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Properties

Label 2-370-185.68-c1-0-6
Degree $2$
Conductor $370$
Sign $0.309 - 0.950i$
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 + 4-s + (−1 + 2i)5-s + (−2 + 2i)7-s + 8-s + 3i·9-s + (−1 + 2i)10-s + 4·13-s + (−2 + 2i)14-s + 16-s + 2i·17-s + 3i·18-s + (−2 − 2i)19-s + (−1 + 2i)20-s + 4·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.447 + 0.894i)5-s + (−0.755 + 0.755i)7-s + 0.353·8-s + i·9-s + (−0.316 + 0.632i)10-s + 1.10·13-s + (−0.534 + 0.534i)14-s + 0.250·16-s + 0.485i·17-s + 0.707i·18-s + (−0.458 − 0.458i)19-s + (−0.223 + 0.447i)20-s + 0.834·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.309 - 0.950i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.309 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42359 + 1.03404i\)
\(L(\frac12)\) \(\approx\) \(1.42359 + 1.03404i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
37 \( 1 + (1 - 6i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + (2 + 2i)T + 19iT^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (-7 + 7i)T - 29iT^{2} \)
31 \( 1 + (4 + 4i)T + 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (2 - 2i)T - 47iT^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (2 + 2i)T + 59iT^{2} \)
61 \( 1 + (-1 - i)T + 61iT^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 + (12 + 12i)T + 79iT^{2} \)
83 \( 1 + (-4 - 4i)T + 83iT^{2} \)
89 \( 1 + (-7 + 7i)T - 89iT^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−11.47718013602246003375742552267, −10.91660126747867709884420843737, −10.01220533968639049782248084203, −8.664288034810394084445726496172, −7.72891915311317816813286844699, −6.55237808437988363901969639400, −5.94187678596989010516857676987, −4.56135296400034752490946096296, −3.34807128733237594577014040058, −2.36705659868046913175965849507, 1.03199153426286123389538234781, 3.34486315504660437022090896820, 4.01037918190529780212636860751, 5.23245744764443935254708726779, 6.41684643507938113959068339258, 7.17433757636585765934880539061, 8.514673371182734911046364075142, 9.281916531279392146854246427040, 10.47355051026309155446020727050, 11.34700688141165288944093319966

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