Properties

Label 2-370-185.68-c1-0-0
Degree $2$
Conductor $370$
Sign $-0.578 - 0.815i$
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 + (1.29 − 1.29i)3-s + 4-s + (−2.20 + 0.390i)5-s + (−1.29 + 1.29i)6-s + (−2.67 + 2.67i)7-s − 8-s − 0.348i·9-s + (2.20 − 0.390i)10-s + 1.29i·11-s + (1.29 − 1.29i)12-s − 6.92·13-s + (2.67 − 2.67i)14-s + (−2.34 + 3.35i)15-s + 16-s + 4.85i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.747 − 0.747i)3-s + 0.5·4-s + (−0.984 + 0.174i)5-s + (−0.528 + 0.528i)6-s + (−1.00 + 1.00i)7-s − 0.353·8-s − 0.116i·9-s + (0.696 − 0.123i)10-s + 0.390i·11-s + (0.373 − 0.373i)12-s − 1.92·13-s + (0.713 − 0.713i)14-s + (−0.604 + 0.866i)15-s + 0.250·16-s + 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.578 - 0.815i$
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: \((2,\ 370,\ (\ :1/2),\ -0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158682 + 0.307190i\)
\(L(\frac12)\) \(\approx\) \(0.158682 + 0.307190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 0.390i)T \)
37 \( 1 + (6.03 - 0.770i)T \)
good3 \( 1 + (-1.29 + 1.29i)T - 3iT^{2} \)
7 \( 1 + (2.67 - 2.67i)T - 7iT^{2} \)
11 \( 1 - 1.29iT - 11T^{2} \)
13 \( 1 + 6.92T + 13T^{2} \)
17 \( 1 - 4.85iT - 17T^{2} \)
19 \( 1 + (4.69 + 4.69i)T + 19iT^{2} \)
23 \( 1 - 3.33T + 23T^{2} \)
29 \( 1 + (-3.73 + 3.73i)T - 29iT^{2} \)
31 \( 1 + (0.146 + 0.146i)T + 31iT^{2} \)
41 \( 1 - 6.38iT - 41T^{2} \)
43 \( 1 + 2.51T + 43T^{2} \)
47 \( 1 + (2.93 - 2.93i)T - 47iT^{2} \)
53 \( 1 + (1.30 + 1.30i)T + 53iT^{2} \)
59 \( 1 + (-4.70 - 4.70i)T + 59iT^{2} \)
61 \( 1 + (7.81 + 7.81i)T + 61iT^{2} \)
67 \( 1 + (-3.10 - 3.10i)T + 67iT^{2} \)
71 \( 1 + 5.59T + 71T^{2} \)
73 \( 1 + (-0.134 + 0.134i)T - 73iT^{2} \)
79 \( 1 + (-10.8 - 10.8i)T + 79iT^{2} \)
83 \( 1 + (8.74 + 8.74i)T + 83iT^{2} \)
89 \( 1 + (-3.83 + 3.83i)T - 89iT^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−11.89932071701319869342548865760, −10.68833981198372664775142537212, −9.690985747006468718291776179314, −8.769959370569853141348326092358, −8.081251813532202067469901951649, −7.15166049354468097997930604618, −6.50269040149944890498138471144, −4.76931144375445609442218723334, −3.01164337525710532415074659915, −2.26712868570617692916936398062, 0.24753968932245616152820951048, 2.88498273281120103866781399386, 3.73906245586255080400870195419, 4.85508606736216611756263390128, 6.78286425022798306615591831970, 7.38181953293403024196672461121, 8.451859588618355487009110115522, 9.293373774187147197675687474942, 10.05881508277880576063205628789, 10.67093791088449964915085463143

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