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Properties

Label 2-370-185.117-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.266 - 0.963i$
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 + (1.82 + 1.82i)3-s + 4-s + (−1.23 + 1.86i)5-s + (−1.82 − 1.82i)6-s + (3.41 + 3.41i)7-s − 8-s + 3.68i·9-s + (1.23 − 1.86i)10-s − 3.97i·11-s + (1.82 + 1.82i)12-s − 4.28·13-s + (−3.41 − 3.41i)14-s + (−5.66 + 1.14i)15-s + 16-s + 2.57i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (1.05 + 1.05i)3-s + 0.5·4-s + (−0.553 + 0.833i)5-s + (−0.746 − 0.746i)6-s + (1.29 + 1.29i)7-s − 0.353·8-s + 1.22i·9-s + (0.391 − 0.589i)10-s − 1.19i·11-s + (0.527 + 0.527i)12-s − 1.18·13-s + (−0.912 − 0.912i)14-s + (−1.46 + 0.295i)15-s + 0.250·16-s + 0.624i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 370,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819811 + 1.07732i\)
\(L(\frac12)\) \(\approx\) \(0.819811 + 1.07732i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 1.86i)T \)
37 \( 1 + (-5.60 - 2.37i)T \)
good3 \( 1 + (-1.82 - 1.82i)T + 3iT^{2} \)
7 \( 1 + (-3.41 - 3.41i)T + 7iT^{2} \)
11 \( 1 + 3.97iT - 11T^{2} \)
13 \( 1 + 4.28T + 13T^{2} \)
17 \( 1 - 2.57iT - 17T^{2} \)
19 \( 1 + (-5.24 + 5.24i)T - 19iT^{2} \)
23 \( 1 + 2.21T + 23T^{2} \)
29 \( 1 + (2.03 + 2.03i)T + 29iT^{2} \)
31 \( 1 + (3.23 - 3.23i)T - 31iT^{2} \)
41 \( 1 - 2.50iT - 41T^{2} \)
43 \( 1 - 9.00T + 43T^{2} \)
47 \( 1 + (0.943 + 0.943i)T + 47iT^{2} \)
53 \( 1 + (-2.84 + 2.84i)T - 53iT^{2} \)
59 \( 1 + (2.91 - 2.91i)T - 59iT^{2} \)
61 \( 1 + (5.71 - 5.71i)T - 61iT^{2} \)
67 \( 1 + (1.28 - 1.28i)T - 67iT^{2} \)
71 \( 1 - 8.40T + 71T^{2} \)
73 \( 1 + (5.32 + 5.32i)T + 73iT^{2} \)
79 \( 1 + (-9.77 + 9.77i)T - 79iT^{2} \)
83 \( 1 + (-2.57 + 2.57i)T - 83iT^{2} \)
89 \( 1 + (9.42 + 9.42i)T + 89iT^{2} \)
97 \( 1 - 5.92iT - 97T^{2} \)
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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

−11.37325271751335867086574900270, −10.72315235006316220661875606185, −9.617370859784670794256856599497, −8.900431390704145168298756459947, −8.165158917558288058093692027839, −7.46810022102659624467773327923, −5.81390651224386532896037562210, −4.60916079999850288439535427837, −3.16733734258044410961269775915, −2.40545221324329982598928972411, 1.12908165864844865392877165503, 2.13775012569130910502189786469, 3.94989483371013661939879726163, 5.09797267899032121610033612598, 7.18228785537843303805835504063, 7.72421173417372108159635400273, 7.82449383150387630615942603749, 9.178598135578866690481900332336, 9.918916666655205564509292130775, 11.20743028255744941113634505297

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