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Properties

Label 2-370-185.142-c1-0-12
Degree $2$
Conductor $370$
Sign $0.148 + 0.988i$
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 − 4-s + (−2 + i)5-s + (−2 − 2i)7-s − i·8-s − 3i·9-s + (−1 − 2i)10-s − 4i·13-s + (2 − 2i)14-s + 16-s − 2·17-s + 3·18-s + (2 + 2i)19-s + (2 − i)20-s − 4i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s + (−0.755 − 0.755i)7-s − 0.353i·8-s − i·9-s + (−0.316 − 0.632i)10-s − 1.10i·13-s + (0.534 − 0.534i)14-s + 0.250·16-s − 0.485·17-s + 0.707·18-s + (0.458 + 0.458i)19-s + (0.447 − 0.223i)20-s − 0.834i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.148 + 0.988i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.148 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.398582 - 0.343129i\)
\(L(\frac12)\) \(\approx\) \(0.398582 - 0.343129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - i)T \)
37 \( 1 + (6 + i)T \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (-2 - 2i)T + 19iT^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + (7 - 7i)T - 29iT^{2} \)
31 \( 1 + (4 + 4i)T + 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
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 \( 1 + 12T + 97T^{2} \)
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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

−11.05837547175172259104115780760, −10.25618407552202437619726152264, −9.258357554722968770592435204692, −8.226410853831396909943080683464, −7.22383955773552856424802437728, −6.68162990850165074635186615510, −5.48336745880939156782373721048, −3.94763464741997591106777175486, −3.31976316621932133557067001860, −0.34922906211715481459469904070, 1.99332978472701014131446014688, 3.38876272255592423234517035955, 4.52095980663305453796274492821, 5.52774474511993154997395398781, 6.99996654914562854651426053592, 8.049025178886690148935666509297, 9.050472398493849135696134044775, 9.623516144900915431462891346675, 11.01953573661147156632064910491, 11.52408095105280412824585195220

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