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Properties

Label 2-370-37.12-c1-0-11
Degree $2$
Conductor $370$
Sign $0.0837 + 0.996i$
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  + (0.766 − 0.642i)2-s + (−0.599 − 0.503i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)5-s − 0.782·6-s + (2.34 − 0.853i)7-s + (−0.500 − 0.866i)8-s + (−0.414 − 2.35i)9-s + (0.5 − 0.866i)10-s + (0.923 + 1.59i)11-s + (−0.599 + 0.503i)12-s + (−0.763 + 4.33i)13-s + (1.24 − 2.16i)14-s + (−0.735 − 0.267i)15-s + (−0.939 − 0.342i)16-s + (−0.974 − 5.52i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.346 − 0.290i)3-s + (0.0868 − 0.492i)4-s + (0.420 − 0.152i)5-s − 0.319·6-s + (0.886 − 0.322i)7-s + (−0.176 − 0.306i)8-s + (−0.138 − 0.783i)9-s + (0.158 − 0.273i)10-s + (0.278 + 0.482i)11-s + (−0.173 + 0.145i)12-s + (−0.211 + 1.20i)13-s + (0.333 − 0.577i)14-s + (−0.189 − 0.0691i)15-s + (−0.234 − 0.0855i)16-s + (−0.236 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0837 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0837 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.0837 + 0.996i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.0837 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32094 - 1.21457i\)
\(L(\frac12)\) \(\approx\) \(1.32094 - 1.21457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (5.76 - 1.93i)T \)
good3 \( 1 + (0.599 + 0.503i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-2.34 + 0.853i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-0.923 - 1.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.763 - 4.33i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.974 + 5.52i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (3.01 + 2.52i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-2.88 + 5.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.06 - 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.55T + 31T^{2} \)
41 \( 1 + (1.37 - 7.78i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 5.64T + 43T^{2} \)
47 \( 1 + (3.47 - 6.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.91 - 1.79i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-10.3 - 3.75i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (2.47 - 14.0i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-3.74 + 1.36i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-7.88 - 6.61i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 2.47T + 73T^{2} \)
79 \( 1 + (-10.9 + 3.98i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (2.30 + 13.0i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (7.26 + 2.64i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (6.64 - 11.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.58586200128834308322479163818, −10.43419256000300314496804262133, −9.428885086233798901131769846353, −8.599615740365364346699910157304, −6.94237871418956482992313114742, −6.51867891017983862549430637592, −4.96354954009323484886904684129, −4.41669486848635734303778070450, −2.66148623764219875003232321501, −1.23353520256776253013532496480, 2.09641785469965382483398529935, 3.67062764630684571639931568328, 5.02982674936090448330004822019, 5.59260901728829574093806067167, 6.61195095537371272962415775101, 8.124140692810120410369973564178, 8.361321140506654752511788078042, 10.03288400480148062416476152981, 10.76556704459059859686812465851, 11.57806963691095447179120342750

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