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Properties

Label 2-34e2-17.13-c1-0-0
Degree $2$
Conductor $1156$
Sign $-0.615 - 0.788i$
Analytic cond. $9.23070$
Root an. cond. $3.03820$
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  + (−1.30 − 1.30i)3-s + (1 + i)5-s + (1.30 − 1.30i)7-s + 0.394i·9-s + (−3.30 + 3.30i)11-s − 4.60·13-s − 2.60i·15-s + 6.60i·19-s − 3.39·21-s + (0.697 − 0.697i)23-s − 3i·25-s + (−3.39 + 3.39i)27-s + (−5.60 − 5.60i)29-s + (−0.697 − 0.697i)31-s + 8.60·33-s + ⋯
L(s)  = 1  + (−0.752 − 0.752i)3-s + (0.447 + 0.447i)5-s + (0.492 − 0.492i)7-s + 0.131i·9-s + (−0.995 + 0.995i)11-s − 1.27·13-s − 0.672i·15-s + 1.51i·19-s − 0.740·21-s + (0.145 − 0.145i)23-s − 0.600i·25-s + (−0.653 + 0.653i)27-s + (−1.04 − 1.04i)29-s + (−0.125 − 0.125i)31-s + 1.49·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ -0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2429313347\)
\(L(\frac12)\) \(\approx\) \(0.2429313347\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (1.30 + 1.30i)T + 3iT^{2} \)
5 \( 1 + (-1 - i)T + 5iT^{2} \)
7 \( 1 + (-1.30 + 1.30i)T - 7iT^{2} \)
11 \( 1 + (3.30 - 3.30i)T - 11iT^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
19 \( 1 - 6.60iT - 19T^{2} \)
23 \( 1 + (-0.697 + 0.697i)T - 23iT^{2} \)
29 \( 1 + (5.60 + 5.60i)T + 29iT^{2} \)
31 \( 1 + (0.697 + 0.697i)T + 31iT^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + (1 - i)T - 41iT^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 9.21iT - 53T^{2} \)
59 \( 1 - 1.39iT - 59T^{2} \)
61 \( 1 + (8.21 - 8.21i)T - 61iT^{2} \)
67 \( 1 + 5.21T + 67T^{2} \)
71 \( 1 + (-7.90 - 7.90i)T + 71iT^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 + (3.30 - 3.30i)T - 79iT^{2} \)
83 \( 1 - 3.81iT - 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + (-0.394 - 0.394i)T + 97iT^{2} \)
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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

−10.01925584826898830136689512588, −9.661425885082467832284634809001, −8.042064518579777783569166019815, −7.53483549558031911926641653184, −6.82240989193575810467041539260, −5.91491182424852798155423273629, −5.14461763997875636607746841691, −4.12711914194622165131483621105, −2.56318663851318911030380002844, −1.61869693217034546398410188641, 0.10852602725224599477152861186, 2.03819320952508562144621833576, 3.20078115881859944642419329123, 4.75116169041185348500482959712, 5.19471052670762930350767803901, 5.65920983190022866918171254482, 6.99505876314903566631623832724, 7.908853480098417911463299690802, 8.920796763006133587521661361456, 9.455991190290940877811680352789

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