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Properties

Label 2-370-5.4-c1-0-6
Degree $2$
Conductor $370$
Sign $0.762 - 0.646i$
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 + 2.62i·3-s − 4-s + (1.70 − 1.44i)5-s + 2.62·6-s + 1.83i·7-s + i·8-s − 3.89·9-s + (−1.44 − 1.70i)10-s + 4.19·11-s − 2.62i·12-s − 0.369i·13-s + 1.83·14-s + (3.79 + 4.47i)15-s + 16-s + 5.08i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.51i·3-s − 0.5·4-s + (0.762 − 0.646i)5-s + 1.07·6-s + 0.692i·7-s + 0.353i·8-s − 1.29·9-s + (−0.457 − 0.539i)10-s + 1.26·11-s − 0.757i·12-s − 0.102i·13-s + 0.489·14-s + (0.980 + 1.15i)15-s + 0.250·16-s + 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.762 - 0.646i$
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: \((2,\ 370,\ (\ :1/2),\ 0.762 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37764 + 0.505616i\)
\(L(\frac12)\) \(\approx\) \(1.37764 + 0.505616i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 1.44i)T \)
37 \( 1 + iT \)
good3 \( 1 - 2.62iT - 3T^{2} \)
7 \( 1 - 1.83iT - 7T^{2} \)
11 \( 1 - 4.19T + 11T^{2} \)
13 \( 1 + 0.369iT - 13T^{2} \)
17 \( 1 - 5.08iT - 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 - 5.62iT - 23T^{2} \)
29 \( 1 + 1.20T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
41 \( 1 + 8.01T + 41T^{2} \)
43 \( 1 + 2.27iT - 43T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + 9.94iT - 53T^{2} \)
59 \( 1 - 5.34T + 59T^{2} \)
61 \( 1 + 9.79T + 61T^{2} \)
67 \( 1 + 1.85iT - 67T^{2} \)
71 \( 1 - 2.86T + 71T^{2} \)
73 \( 1 + 8.09iT - 73T^{2} \)
79 \( 1 + 6.06T + 79T^{2} \)
83 \( 1 + 8.93iT - 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + 6.05iT - 97T^{2} \)
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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

−11.45242072300559178676905444079, −10.27921196098941964021646622906, −9.878054097338939385932952051973, −8.892122956925646261534946046606, −8.550835254716635621981619987209, −6.29396687202998277697633103484, −5.36560315906133467785132791607, −4.39486599754975174503097917282, −3.49608328756568861810200040506, −1.84213289369450174461901842888, 1.15742013890095378258044901024, 2.68946954117892575433683834986, 4.45030878689047533762249394042, 6.06058979012472761341266422193, 6.67693257951949566975457504481, 7.15177947766005251355371343765, 8.238062434702185586076647911974, 9.269728557592769380594043836003, 10.30885637107653620888659353485, 11.47529982634392767198273018490

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