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Properties

Label 2-155-155.92-c1-0-6
Degree $2$
Conductor $155$
Sign $0.870 + 0.492i$
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  + (−0.759 − 0.759i)2-s + (1.72 + 1.72i)3-s − 0.845i·4-s + (1.60 − 1.55i)5-s − 2.62i·6-s + (−0.240 − 0.240i)7-s + (−2.16 + 2.16i)8-s + 2.95i·9-s + (−2.40 − 0.0370i)10-s − 1.92i·11-s + (1.45 − 1.45i)12-s + (2.27 + 2.27i)13-s + 0.365i·14-s + (5.45 + 0.0842i)15-s + 1.59·16-s + (−1.37 + 1.37i)17-s + ⋯
L(s)  = 1  + (−0.537 − 0.537i)2-s + (0.995 + 0.995i)3-s − 0.422i·4-s + (0.717 − 0.696i)5-s − 1.07i·6-s + (−0.0908 − 0.0908i)7-s + (−0.764 + 0.764i)8-s + 0.983i·9-s + (−0.759 − 0.0117i)10-s − 0.578i·11-s + (0.421 − 0.421i)12-s + (0.629 + 0.629i)13-s + 0.0975i·14-s + (1.40 + 0.0217i)15-s + 0.398·16-s + (−0.333 + 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.870 + 0.492i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.870 + 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18193 - 0.310985i\)
\(L(\frac12)\) \(\approx\) \(1.18193 - 0.310985i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.60 + 1.55i)T \)
31 \( 1 + (1.47 - 5.37i)T \)
good2 \( 1 + (0.759 + 0.759i)T + 2iT^{2} \)
3 \( 1 + (-1.72 - 1.72i)T + 3iT^{2} \)
7 \( 1 + (0.240 + 0.240i)T + 7iT^{2} \)
11 \( 1 + 1.92iT - 11T^{2} \)
13 \( 1 + (-2.27 - 2.27i)T + 13iT^{2} \)
17 \( 1 + (1.37 - 1.37i)T - 17iT^{2} \)
19 \( 1 - 1.20iT - 19T^{2} \)
23 \( 1 + (0.350 + 0.350i)T + 23iT^{2} \)
29 \( 1 + 8.98T + 29T^{2} \)
37 \( 1 + (5.45 - 5.45i)T - 37iT^{2} \)
41 \( 1 + 7.24T + 41T^{2} \)
43 \( 1 + (-8.07 - 8.07i)T + 43iT^{2} \)
47 \( 1 + (0.480 + 0.480i)T + 47iT^{2} \)
53 \( 1 + (7.72 + 7.72i)T + 53iT^{2} \)
59 \( 1 + 5.59iT - 59T^{2} \)
61 \( 1 - 0.168iT - 61T^{2} \)
67 \( 1 + (-9.09 - 9.09i)T + 67iT^{2} \)
71 \( 1 + 3.31T + 71T^{2} \)
73 \( 1 + (6.63 + 6.63i)T + 73iT^{2} \)
79 \( 1 - 6.23T + 79T^{2} \)
83 \( 1 + (5.08 + 5.08i)T + 83iT^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + (4.74 + 4.74i)T + 97iT^{2} \)
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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

−13.03829873686188016762720497059, −11.53294181173057685858508794754, −10.49916401038798096370959873394, −9.727348135289119136996369161576, −8.954615030478322444597511945463, −8.425242191628836346585930143603, −6.23350666296067255452562749289, −4.97213915361083356062256642895, −3.48698005608985461096600785760, −1.82748243446157230947833873301, 2.20267175893045183040178806551, 3.44968795717952372504279986880, 5.94101838254348034182531505576, 7.11727333994823988024097925865, 7.60367921182289933788574024038, 8.802461100038739695096210171544, 9.516652720605429313753118089917, 10.92931158544671462477465837146, 12.42088511994595674059060563373, 13.16502654977931989313400477570

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