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Properties

Label 2-370-185.49-c1-0-15
Degree $2$
Conductor $370$
Sign $0.640 + 0.768i$
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  + (0.642 + 0.766i)2-s + (0.958 − 1.14i)3-s + (−0.173 + 0.984i)4-s + (−0.774 − 2.09i)5-s + 1.49·6-s + (−1.14 − 3.14i)7-s + (−0.866 + 0.500i)8-s + (0.134 + 0.764i)9-s + (1.10 − 1.94i)10-s + (−2.82 − 4.89i)11-s + (0.958 + 1.14i)12-s + (7.01 + 1.23i)13-s + (1.67 − 2.89i)14-s + (−3.13 − 1.12i)15-s + (−0.939 − 0.342i)16-s + (0.135 − 0.0239i)17-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (0.553 − 0.659i)3-s + (−0.0868 + 0.492i)4-s + (−0.346 − 0.938i)5-s + 0.608·6-s + (−0.432 − 1.18i)7-s + (−0.306 + 0.176i)8-s + (0.0449 + 0.254i)9-s + (0.350 − 0.614i)10-s + (−0.852 − 1.47i)11-s + (0.276 + 0.329i)12-s + (1.94 + 0.343i)13-s + (0.446 − 0.773i)14-s + (−0.810 − 0.290i)15-s + (−0.234 − 0.0855i)16-s + (0.0329 − 0.00581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.640 + 0.768i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.640 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62475 - 0.760811i\)
\(L(\frac12)\) \(\approx\) \(1.62475 - 0.760811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 + (0.774 + 2.09i)T \)
37 \( 1 + (-3.63 + 4.87i)T \)
good3 \( 1 + (-0.958 + 1.14i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.14 + 3.14i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.82 + 4.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-7.01 - 1.23i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.135 + 0.0239i)T + (15.9 - 5.81i)T^{2} \)
19 \( 1 + (-2.40 - 2.01i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (4.10 + 2.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.66 - 6.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.49T + 31T^{2} \)
41 \( 1 + (-0.125 + 0.712i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 + (-2.72 - 1.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.74 + 4.79i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.05 + 0.383i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.68 - 9.56i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (3.86 + 10.6i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-9.39 - 7.88i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 1.50iT - 73T^{2} \)
79 \( 1 + (-2.72 + 0.993i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (11.8 - 2.08i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-6.23 - 2.26i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-3.46 - 1.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.28564351228051750928398926372, −10.53627992032524335019076006536, −8.962703555349037842193921371955, −8.208933101333055274832184496445, −7.71832601498623273413968328290, −6.49789055477330740465552620075, −5.52324199235304283376423211857, −4.13968266567379422398920600976, −3.25823598995924571224746205195, −1.10149501637504228749281926298, 2.36116237646833587407833820696, 3.28362605569272106661472319192, 4.19744490126583031290211093651, 5.63525992278287171098070264533, 6.55617169906539315916844431523, 7.931200410272519653407414580813, 9.001430458598950551858363051133, 9.880843546534300578777924481726, 10.49468709435372321320011726847, 11.59930512706848124457380895122

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