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Properties

Label 2-370-37.21-c1-0-13
Degree $2$
Conductor $370$
Sign $-0.000155 + 0.999i$
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  + (0.342 − 0.939i)2-s + (3.03 − 1.10i)3-s + (−0.766 − 0.642i)4-s + (−0.984 − 0.173i)5-s − 3.23i·6-s + (0.130 − 0.739i)7-s + (−0.866 + 0.500i)8-s + (5.70 − 4.78i)9-s + (−0.5 + 0.866i)10-s + (1.20 + 2.08i)11-s + (−3.03 − 1.10i)12-s + (−2.83 + 3.37i)13-s + (−0.650 − 0.375i)14-s + (−3.18 + 0.561i)15-s + (0.173 + 0.984i)16-s + (−0.673 − 0.803i)17-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (1.75 − 0.638i)3-s + (−0.383 − 0.321i)4-s + (−0.440 − 0.0776i)5-s − 1.31i·6-s + (0.0492 − 0.279i)7-s + (−0.306 + 0.176i)8-s + (1.90 − 1.59i)9-s + (−0.158 + 0.273i)10-s + (0.363 + 0.628i)11-s + (−0.876 − 0.319i)12-s + (−0.784 + 0.935i)13-s + (−0.173 − 0.100i)14-s + (−0.821 + 0.144i)15-s + (0.0434 + 0.246i)16-s + (−0.163 − 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000155 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.000155 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.000155 + 0.999i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.000155 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65181 - 1.65207i\)
\(L(\frac12)\) \(\approx\) \(1.65181 - 1.65207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
5 \( 1 + (0.984 + 0.173i)T \)
37 \( 1 + (-5.29 + 2.99i)T \)
good3 \( 1 + (-3.03 + 1.10i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.130 + 0.739i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-1.20 - 2.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.83 - 3.37i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.673 + 0.803i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (1.37 + 3.78i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (-2.42 - 1.40i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.43 - 3.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.5iT - 31T^{2} \)
41 \( 1 + (6.67 + 5.59i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 6.19iT - 43T^{2} \)
47 \( 1 + (-1.01 + 1.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.313 - 1.77i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-5.19 + 0.916i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-6.70 + 7.98i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.144 - 0.820i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (14.1 - 5.15i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 2.93T + 73T^{2} \)
79 \( 1 + (7.33 + 1.29i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.30 + 2.77i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (13.8 - 2.43i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (-3.56 - 2.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.34207348682325376245377493114, −10.03981251870544168212313889430, −9.149962203642225313137040126739, −8.672424516974074227739199346710, −7.30799408686354881782644381400, −6.94990927770477433062854290329, −4.72888112997376833318795747664, −3.80138245452446769701476272020, −2.69884218255453287285036696783, −1.59568043697594910680473416229, 2.50732300401944597452733146494, 3.58477055598346365970824975802, 4.40714072176432070206307398827, 5.75082969928043520198765973238, 7.28882188846954998085426961735, 8.037829238447073534527240462939, 8.617136327553931455590084274987, 9.564750896622276423249593309503, 10.34597313185570058924709313858, 11.69409276457493748180815893237

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