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Properties

Label 2-155-31.16-c1-0-8
Degree $2$
Conductor $155$
Sign $0.940 + 0.340i$
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  + (1.16 − 0.843i)2-s + (0.518 + 0.376i)3-s + (0.0178 − 0.0550i)4-s + 5-s + 0.919·6-s + (0.0708 − 0.218i)7-s + (0.860 + 2.64i)8-s + (−0.800 − 2.46i)9-s + (1.16 − 0.843i)10-s + (−0.100 + 0.309i)11-s + (0.0299 − 0.0217i)12-s + (−3.12 − 2.27i)13-s + (−0.101 − 0.312i)14-s + (0.518 + 0.376i)15-s + (3.32 + 2.41i)16-s + (−0.522 − 1.60i)17-s + ⋯
L(s)  = 1  + (0.820 − 0.596i)2-s + (0.299 + 0.217i)3-s + (0.00893 − 0.0275i)4-s + 0.447·5-s + 0.375·6-s + (0.0267 − 0.0824i)7-s + (0.304 + 0.936i)8-s + (−0.266 − 0.820i)9-s + (0.366 − 0.266i)10-s + (−0.0303 + 0.0934i)11-s + (0.00865 − 0.00629i)12-s + (−0.866 − 0.629i)13-s + (−0.0271 − 0.0836i)14-s + (0.133 + 0.0972i)15-s + (0.831 + 0.604i)16-s + (−0.126 − 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.940 + 0.340i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.940 + 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74838 - 0.306932i\)
\(L(\frac12)\) \(\approx\) \(1.74838 - 0.306932i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
31 \( 1 + (-5.37 + 1.46i)T \)
good2 \( 1 + (-1.16 + 0.843i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.518 - 0.376i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.0708 + 0.218i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.100 - 0.309i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (3.12 + 2.27i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.522 + 1.60i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.95 - 3.59i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.197 + 0.606i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.68 - 4.13i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 2.11T + 37T^{2} \)
41 \( 1 + (3.34 - 2.43i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-3.03 + 2.20i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-5.92 - 4.30i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.92 + 5.93i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.91 + 2.84i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 5.36T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + (1.64 + 5.06i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.40 + 13.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.37 - 13.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.71 + 3.42i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.69 + 5.21i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.594 - 1.83i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.72240829673376058794428791147, −12.23241733307388487827486094094, −11.03664172881011330546809161347, −9.995855618444944314066630537175, −8.888593871262721597974048679707, −7.73968559091228325952752453785, −6.15217244645454402383342962170, −4.88172303442728736730566169561, −3.65494119367335118091914231102, −2.42670264114496618428889364951, 2.29899053107781891555029852892, 4.29346952214417533378587335526, 5.33679575122734240778591752398, 6.45549649145092058024231645468, 7.48391281706804612825631715897, 8.813952404644474920339022465640, 9.952933712962494244481551225798, 11.00628270643383799635987820518, 12.41123567349010215500234549916, 13.36928794359996664945449800068

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