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Properties

Label 2-155-155.68-c1-0-7
Degree $2$
Conductor $155$
Sign $0.713 - 0.700i$
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  + (−0.0988 + 0.0988i)2-s + (2.41 + 0.645i)3-s + 1.98i·4-s + (−2.03 − 0.931i)5-s + (−0.302 + 0.174i)6-s + (1.97 + 0.530i)7-s + (−0.393 − 0.393i)8-s + (2.79 + 1.61i)9-s + (0.292 − 0.108i)10-s + (−2.20 − 1.27i)11-s + (−1.27 + 4.77i)12-s + (0.123 + 0.461i)13-s + (−0.247 + 0.143i)14-s + (−4.29 − 3.55i)15-s − 3.88·16-s + (1.29 − 4.82i)17-s + ⋯
L(s)  = 1  + (−0.0698 + 0.0698i)2-s + (1.39 + 0.372i)3-s + 0.990i·4-s + (−0.909 − 0.416i)5-s + (−0.123 + 0.0711i)6-s + (0.748 + 0.200i)7-s + (−0.139 − 0.139i)8-s + (0.931 + 0.537i)9-s + (0.0926 − 0.0344i)10-s + (−0.664 − 0.383i)11-s + (−0.369 + 1.37i)12-s + (0.0342 + 0.127i)13-s + (−0.0662 + 0.0382i)14-s + (−1.10 − 0.918i)15-s − 0.970·16-s + (0.313 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.713 - 0.700i$
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: \((2,\ 155,\ (\ :1/2),\ 0.713 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34806 + 0.550949i\)
\(L(\frac12)\) \(\approx\) \(1.34806 + 0.550949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.03 + 0.931i)T \)
31 \( 1 + (-0.314 + 5.55i)T \)
good2 \( 1 + (0.0988 - 0.0988i)T - 2iT^{2} \)
3 \( 1 + (-2.41 - 0.645i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.97 - 0.530i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.20 + 1.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.123 - 0.461i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.29 + 4.82i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.05 + 2.34i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.82 - 3.82i)T - 23iT^{2} \)
29 \( 1 - 2.03T + 29T^{2} \)
37 \( 1 + (0.00153 - 0.00573i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.31 + 4.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (11.1 + 3.00i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (7.40 - 7.40i)T - 47iT^{2} \)
53 \( 1 + (-2.50 - 9.33i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.796 + 0.459i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.4iT - 61T^{2} \)
67 \( 1 + (-3.38 - 12.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.54 + 7.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.50 + 5.61i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.28 - 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.58 - 5.91i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (-0.235 + 0.235i)T - 97iT^{2} \)
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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

−13.31630730019030192753587762899, −11.97700965206091871431637083027, −11.37296473562281607168564960702, −9.613155460904689763301418061590, −8.723153988258336264252590953477, −7.946933517609758421571746850499, −7.44677434619529467158912167915, −4.96240424973262286537063292761, −3.75678088190762712526683819899, −2.74332254621338632851700127819, 1.84224344153123909616703633847, 3.40719863224326843097715062466, 4.91342916440812172450625880686, 6.65807972624378880000136538790, 7.967003422296695922608736046753, 8.319628224869902679853306540949, 9.852376278014945965441968231125, 10.64133172847518113855517686752, 11.79961709130447601308078506642, 13.01942575364128180686534064041

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