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Properties

Label 2-370-185.117-c1-0-5
Degree $2$
Conductor $370$
Sign $0.376 - 0.926i$
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 + (1.28 + 1.28i)3-s + 4-s + (1.69 + 1.45i)5-s + (−1.28 − 1.28i)6-s + (0.579 + 0.579i)7-s − 8-s + 0.324i·9-s + (−1.69 − 1.45i)10-s + 3.64i·11-s + (1.28 + 1.28i)12-s + 3.10·13-s + (−0.579 − 0.579i)14-s + (0.301 + 4.06i)15-s + 16-s − 8.09i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.744 + 0.744i)3-s + 0.5·4-s + (0.757 + 0.652i)5-s + (−0.526 − 0.526i)6-s + (0.219 + 0.219i)7-s − 0.353·8-s + 0.108i·9-s + (−0.535 − 0.461i)10-s + 1.09i·11-s + (0.372 + 0.372i)12-s + 0.861·13-s + (−0.154 − 0.154i)14-s + (0.0777 + 1.04i)15-s + 0.250·16-s − 1.96i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18565 + 0.798185i\)
\(L(\frac12)\) \(\approx\) \(1.18565 + 0.798185i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 1.45i)T \)
37 \( 1 + (4.87 - 3.63i)T \)
good3 \( 1 + (-1.28 - 1.28i)T + 3iT^{2} \)
7 \( 1 + (-0.579 - 0.579i)T + 7iT^{2} \)
11 \( 1 - 3.64iT - 11T^{2} \)
13 \( 1 - 3.10T + 13T^{2} \)
17 \( 1 + 8.09iT - 17T^{2} \)
19 \( 1 + (3.05 - 3.05i)T - 19iT^{2} \)
23 \( 1 + 7.79T + 23T^{2} \)
29 \( 1 + (-1.37 - 1.37i)T + 29iT^{2} \)
31 \( 1 + (-3.23 + 3.23i)T - 31iT^{2} \)
41 \( 1 - 9.31iT - 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + (4.11 + 4.11i)T + 47iT^{2} \)
53 \( 1 + (-0.446 + 0.446i)T - 53iT^{2} \)
59 \( 1 + (6.16 - 6.16i)T - 59iT^{2} \)
61 \( 1 + (-8.71 + 8.71i)T - 61iT^{2} \)
67 \( 1 + (2.01 - 2.01i)T - 67iT^{2} \)
71 \( 1 - 0.00151T + 71T^{2} \)
73 \( 1 + (9.32 + 9.32i)T + 73iT^{2} \)
79 \( 1 + (-0.760 + 0.760i)T - 79iT^{2} \)
83 \( 1 + (-3.16 + 3.16i)T - 83iT^{2} \)
89 \( 1 + (5.80 + 5.80i)T + 89iT^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−11.35637349903178541967792769525, −10.12872015520957655727483553024, −9.867030782584455283647560007711, −9.007341150442132330855321480987, −8.060932753544573902148772837212, −6.94109526641169660255623601579, −5.96223810959832223515934673858, −4.48044463051748814040576147632, −3.10976968900083475850239133301, −2.01211419774416913897158231476, 1.29472393702820553404810635433, 2.32921248830963221834218122604, 3.96010636596146139728493430406, 5.76587962309845254524752235535, 6.46411121868357387533082076327, 7.893218569812336127170815288549, 8.486093263931210028921060905936, 8.965175415954867260264122813232, 10.38950949640281381922779247798, 10.88535652022666594780323931264

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