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Properties

Label 2-155-31.8-c1-0-3
Degree $2$
Conductor $155$
Sign $-0.179 - 0.983i$
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  + (−0.339 + 1.04i)2-s + (0.409 + 1.26i)3-s + (0.640 + 0.465i)4-s + 5-s − 1.45·6-s + (−0.354 − 0.257i)7-s + (−2.48 + 1.80i)8-s + (1.00 − 0.729i)9-s + (−0.339 + 1.04i)10-s + (0.912 + 0.663i)11-s + (−0.324 + 0.998i)12-s + (−1.86 − 5.73i)13-s + (0.389 − 0.282i)14-s + (0.409 + 1.26i)15-s + (−0.552 − 1.70i)16-s + (−6.19 + 4.50i)17-s + ⋯
L(s)  = 1  + (−0.240 + 0.739i)2-s + (0.236 + 0.728i)3-s + (0.320 + 0.232i)4-s + 0.447·5-s − 0.595·6-s + (−0.133 − 0.0972i)7-s + (−0.877 + 0.637i)8-s + (0.334 − 0.243i)9-s + (−0.107 + 0.330i)10-s + (0.275 + 0.199i)11-s + (−0.0936 + 0.288i)12-s + (−0.517 − 1.59i)13-s + (0.104 − 0.0755i)14-s + (0.105 + 0.325i)15-s + (−0.138 − 0.425i)16-s + (−1.50 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.179 - 0.983i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ -0.179 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781640 + 0.936855i\)
\(L(\frac12)\) \(\approx\) \(0.781640 + 0.936855i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.951 + 5.48i)T \)
good2 \( 1 + (0.339 - 1.04i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.409 - 1.26i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.354 + 0.257i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.912 - 0.663i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.86 + 5.73i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (6.19 - 4.50i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.762 - 2.34i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.77 + 4.19i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.72 + 5.31i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 3.06T + 37T^{2} \)
41 \( 1 + (0.957 - 2.94i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-2.16 + 6.67i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-2.55 - 7.85i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.09 - 4.42i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.79 - 8.58i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 3.17T + 61T^{2} \)
67 \( 1 + 4.93T + 67T^{2} \)
71 \( 1 + (2.79 - 2.03i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.71 - 4.15i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (8.38 - 6.08i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.19 + 3.69i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (11.6 + 8.43i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.361 + 0.262i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.14893464653132538286185346042, −12.42872512048979388086256472301, −10.95175499974628886047345544907, −10.10434130354445808058892509863, −9.024489422239963007299206926390, −8.075711826195109805774283162576, −6.83958740838208390403511674937, −5.86586566915462852825457856186, −4.31940564629983897115047352966, −2.73391958694578853698699456373, 1.61241958293832910605269639835, 2.73299785944265550124433656516, 4.79247806425176428485442879631, 6.67048149533438551209819664426, 7.00665190145437465624578670803, 8.960677229838679718004721254491, 9.509116804895075560999789634048, 10.86761073711486338717642552420, 11.57975644746251364191317018426, 12.60130678331255922338368367829

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