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Properties

Label 2-34e2-17.13-c1-0-12
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 − i)3-s + (2 + 2i)5-s + (3 − 3i)7-s − i·9-s + (−1 + i)11-s + 4·13-s − 4i·15-s − 4i·19-s − 6·21-s + (−1 + i)23-s + 3i·25-s + (−4 + 4i)27-s + (2 + 2i)29-s + (−3 − 3i)31-s + 2·33-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (0.894 + 0.894i)5-s + (1.13 − 1.13i)7-s − 0.333i·9-s + (−0.301 + 0.301i)11-s + 1.10·13-s − 1.03i·15-s − 0.917i·19-s − 1.30·21-s + (−0.208 + 0.208i)23-s + 0.600i·25-s + (−0.769 + 0.769i)27-s + (0.371 + 0.371i)29-s + (−0.538 − 0.538i)31-s + 0.348·33-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

\(L(1)\) \(\approx\) \(1.820500559\)
\(L(\frac12)\) \(\approx\) \(1.820500559\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + i)T + 3iT^{2} \)
5 \( 1 + (-2 - 2i)T + 5iT^{2} \)
7 \( 1 + (-3 + 3i)T - 7iT^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
13 \( 1 - 4T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + (-2 - 2i)T + 29iT^{2} \)
31 \( 1 + (3 + 3i)T + 31iT^{2} \)
37 \( 1 + (-6 - 6i)T + 37iT^{2} \)
41 \( 1 + (8 - 8i)T - 41iT^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-6 + 6i)T - 61iT^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (5 + 5i)T + 71iT^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + (-3 + 3i)T - 79iT^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−9.931979552080769411439042762986, −8.828052875756047015576655678111, −7.81066282092093893267839798305, −7.01820946134305332471348993384, −6.46257568915487141673421575032, −5.60381847827298744649913725524, −4.55888506322955564745696996036, −3.41642850413149127535818034084, −2.01937244434487226952525971886, −0.984770350853152802725314585194, 1.40500415511239885629965112496, 2.36729674606395693519591252536, 4.06980698066778627151158442849, 4.99009607468445781029915441357, 5.69746051974915816635084096464, 5.93644729030465536165531618144, 7.67520804056004728719674492580, 8.563288158967104184643713485430, 8.920610599311070001035724528878, 9.977266457117249476514702139060

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