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Properties

Label 2-370-37.27-c1-0-5
Degree $2$
Conductor $370$
Sign $0.146 + 0.989i$
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  + (−0.866 − 0.5i)2-s + (−0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + 1.73i·6-s + (1 + 1.73i)7-s − 0.999i·8-s − 0.999·10-s + 5.46·11-s + (0.866 − 1.49i)12-s + (−0.866 + 0.5i)13-s − 1.99i·14-s + (−1.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (1.73 − i)19-s + (0.866 + 0.499i)20-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + 0.707i·6-s + (0.377 + 0.654i)7-s − 0.353i·8-s − 0.316·10-s + 1.64·11-s + (0.250 − 0.433i)12-s + (−0.240 + 0.138i)13-s − 0.534i·14-s + (−0.387 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.397 − 0.229i)19-s + (0.193 + 0.111i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.146 + 0.989i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.146 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773231 - 0.667101i\)
\(L(\frac12)\) \(\approx\) \(0.773231 - 0.667101i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (6.06 + 0.5i)T \)
good3 \( 1 + (0.866 + 1.5i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.46T + 11T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.73 + i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.46iT - 23T^{2} \)
29 \( 1 - 0.535iT - 29T^{2} \)
31 \( 1 + 2.26iT - 31T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 7.92iT - 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + (2.59 - 4.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.46 - 2i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.46 - 3.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.73 - 9.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.46T + 73T^{2} \)
79 \( 1 + (3 - 1.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.26 + 7.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.92 - 4i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.92iT - 97T^{2} \)
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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

−11.44945428596532394388016513408, −10.27174739292491252753926057674, −9.161953244128534407584037815996, −8.687912600990379035153618778246, −7.29298679907247890969640831212, −6.57923784371057094640366166827, −5.58944656747658198271444944586, −4.03982223024861026039479772306, −2.23793044652720837979118737590, −1.07679203176650816575375594695, 1.50620310225230971187876536333, 3.67422605828793524993035547406, 4.78868192794662394033183142343, 5.85124107676889981100956137911, 6.89967061345871209793607213911, 7.80847589949252843462671563237, 9.165541438970766702134971967536, 9.719657908977553441225533482336, 10.60278448685450622425326543829, 11.26125571365052368020960564674

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