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Properties

Label 2-155-155.68-c1-0-9
Degree $2$
Conductor $155$
Sign $0.790 + 0.612i$
Analytic cond. $1.23768$
Root an. cond. $1.11251$
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  + (0.794 − 0.794i)2-s + (−0.192 − 0.0515i)3-s + 0.737i·4-s + (0.941 − 2.02i)5-s + (−0.193 + 0.111i)6-s + (2.34 + 0.629i)7-s + (2.17 + 2.17i)8-s + (−2.56 − 1.48i)9-s + (−0.863 − 2.35i)10-s + (0.369 + 0.213i)11-s + (0.0379 − 0.141i)12-s + (−1.23 − 4.61i)13-s + (2.36 − 1.36i)14-s + (−0.285 + 0.341i)15-s + 1.98·16-s + (−1.93 + 7.23i)17-s + ⋯
L(s)  = 1  + (0.561 − 0.561i)2-s + (−0.111 − 0.0297i)3-s + 0.368i·4-s + (0.421 − 0.906i)5-s + (−0.0790 + 0.0456i)6-s + (0.887 + 0.237i)7-s + (0.768 + 0.768i)8-s + (−0.854 − 0.493i)9-s + (−0.272 − 0.746i)10-s + (0.111 + 0.0643i)11-s + (0.0109 − 0.0409i)12-s + (−0.343 − 1.28i)13-s + (0.632 − 0.365i)14-s + (−0.0737 + 0.0881i)15-s + 0.495·16-s + (−0.470 + 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45203 - 0.497113i\)
\(L(\frac12)\) \(\approx\) \(1.45203 - 0.497113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.941 + 2.02i)T \)
31 \( 1 + (0.465 + 5.54i)T \)
good2 \( 1 + (-0.794 + 0.794i)T - 2iT^{2} \)
3 \( 1 + (0.192 + 0.0515i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.34 - 0.629i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.369 - 0.213i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.23 + 4.61i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.93 - 7.23i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.0296 - 0.0170i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.61 - 5.61i)T - 23iT^{2} \)
29 \( 1 - 1.61T + 29T^{2} \)
37 \( 1 + (1.02 - 3.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.13 - 7.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.580 + 0.155i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-6.32 + 6.32i)T - 47iT^{2} \)
53 \( 1 + (0.290 + 1.08i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-8.63 + 4.98i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 + (-0.862 - 3.21i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.97 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.47 - 9.22i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.20 + 5.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.46 - 5.45i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 7.28T + 89T^{2} \)
97 \( 1 + (-7.17 + 7.17i)T - 97iT^{2} \)
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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

−12.75150881505197789654629924184, −11.98243392027074430807233562370, −11.19935206336828294675790802961, −9.950028182566008633018557516739, −8.414784720984846739559418073957, −8.074583304880627159213743486092, −5.93867002407475832474698661759, −5.01860361956398127703052458639, −3.69250583217657324576454954185, −1.97586583213875101042606059601, 2.28946507947649052801071979844, 4.41047529952521428677809409764, 5.41857351646314309662606367298, 6.59865586046575416614395564698, 7.39781764344147870653563068238, 8.951522847083389091701440222614, 10.21341721773148726875332906338, 11.05643125633749887313732476556, 11.90446728747112758629266533473, 13.72565808577217874156835176083

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