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Properties

Label 2-370-37.34-c1-0-5
Degree $2$
Conductor $370$
Sign $0.942 - 0.332i$
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 + (0.203 − 0.170i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)5-s − 0.265·6-s + (1.72 + 0.628i)7-s + (0.500 − 0.866i)8-s + (−0.508 + 2.88i)9-s + (−0.5 − 0.866i)10-s + (−1.26 + 2.18i)11-s + (0.203 + 0.170i)12-s + (0.483 + 2.74i)13-s + (−0.918 − 1.59i)14-s + (0.249 − 0.0906i)15-s + (−0.939 + 0.342i)16-s + (0.786 − 4.45i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.117 − 0.0983i)3-s + (0.0868 + 0.492i)4-s + (0.420 + 0.152i)5-s − 0.108·6-s + (0.652 + 0.237i)7-s + (0.176 − 0.306i)8-s + (−0.169 + 0.961i)9-s + (−0.158 − 0.273i)10-s + (−0.381 + 0.660i)11-s + (0.0586 + 0.0491i)12-s + (0.134 + 0.761i)13-s + (−0.245 − 0.425i)14-s + (0.0643 − 0.0234i)15-s + (−0.234 + 0.0855i)16-s + (0.190 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.942 - 0.332i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.942 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15987 + 0.198699i\)
\(L(\frac12)\) \(\approx\) \(1.15987 + 0.198699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.599 - 6.05i)T \)
good3 \( 1 + (-0.203 + 0.170i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.72 - 0.628i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.26 - 2.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.483 - 2.74i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.786 + 4.45i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (1.19 - 1.00i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-3.64 - 6.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.63 + 6.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.29T + 31T^{2} \)
41 \( 1 + (0.798 + 4.53i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 + (6.06 + 10.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.0 + 4.02i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (13.1 - 4.77i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.624 - 3.54i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (6.15 + 2.23i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (5.94 - 4.99i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 2.00T + 73T^{2} \)
79 \( 1 + (7.63 + 2.77i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.0889 - 0.504i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-7.93 + 2.88i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.127 - 0.221i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.44643248115803522697874192422, −10.48355506186159437847125262669, −9.680829579652933363633065929481, −8.753357537236598324318289359895, −7.78779717810108562176035522741, −7.01673731640453524133249339640, −5.49009679050595440985297895202, −4.49681327847122667169664547656, −2.73886447456201413374108909051, −1.74669675684761851819543751475, 1.04646393381960816170620820760, 2.93708923047379590311583303650, 4.51454735121760254526201447777, 5.75115393929435386743099857227, 6.50376302673587038561777042749, 7.80811468005685647695811254781, 8.583772442070591605424939668990, 9.283520678371769186672479706393, 10.59363465051563114835047123108, 10.85642449626386883827012676032

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