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Properties

Label 2-1160-145.12-c1-0-0
Degree $2$
Conductor $1160$
Sign $-0.994 + 0.100i$
Analytic cond. $9.26264$
Root an. cond. $3.04345$
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  + 0.480·3-s + (−0.0382 + 2.23i)5-s + (0.312 + 0.312i)7-s − 2.76·9-s + (−1.55 − 1.55i)11-s + (−3.47 − 3.47i)13-s + (−0.0183 + 1.07i)15-s + 3.42i·17-s + (−2.53 + 2.53i)19-s + (0.150 + 0.150i)21-s + (−0.439 + 0.439i)23-s + (−4.99 − 0.171i)25-s − 2.77·27-s + (−4.36 − 3.15i)29-s + (1.56 + 1.56i)31-s + ⋯
L(s)  = 1  + 0.277·3-s + (−0.0171 + 0.999i)5-s + (0.118 + 0.118i)7-s − 0.923·9-s + (−0.468 − 0.468i)11-s + (−0.963 − 0.963i)13-s + (−0.00474 + 0.277i)15-s + 0.831i·17-s + (−0.582 + 0.582i)19-s + (0.0327 + 0.0327i)21-s + (−0.0916 + 0.0916i)23-s + (−0.999 − 0.0342i)25-s − 0.533·27-s + (−0.810 − 0.585i)29-s + (0.281 + 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1160\)    =    \(2^{3} \cdot 5 \cdot 29\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(9.26264\)
Root analytic conductor: \(3.04345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1160} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1160,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2452523698\)
\(L(\frac12)\) \(\approx\) \(0.2452523698\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.0382 - 2.23i)T \)
29 \( 1 + (4.36 + 3.15i)T \)
good3 \( 1 - 0.480T + 3T^{2} \)
7 \( 1 + (-0.312 - 0.312i)T + 7iT^{2} \)
11 \( 1 + (1.55 + 1.55i)T + 11iT^{2} \)
13 \( 1 + (3.47 + 3.47i)T + 13iT^{2} \)
17 \( 1 - 3.42iT - 17T^{2} \)
19 \( 1 + (2.53 - 2.53i)T - 19iT^{2} \)
23 \( 1 + (0.439 - 0.439i)T - 23iT^{2} \)
31 \( 1 + (-1.56 - 1.56i)T + 31iT^{2} \)
37 \( 1 + 0.0934T + 37T^{2} \)
41 \( 1 + (-0.878 + 0.878i)T - 41iT^{2} \)
43 \( 1 - 2.35T + 43T^{2} \)
47 \( 1 + 8.97T + 47T^{2} \)
53 \( 1 + (-0.348 + 0.348i)T - 53iT^{2} \)
59 \( 1 - 0.108iT - 59T^{2} \)
61 \( 1 + (3.28 + 3.28i)T + 61iT^{2} \)
67 \( 1 + (-8.07 + 8.07i)T - 67iT^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + 2.92iT - 73T^{2} \)
79 \( 1 + (11.1 - 11.1i)T - 79iT^{2} \)
83 \( 1 + (8.68 - 8.68i)T - 83iT^{2} \)
89 \( 1 + (-0.0205 + 0.0205i)T - 89iT^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−10.25964341867379457306460284112, −9.521501319394967914858586024216, −8.185493386187909791465954764950, −8.070551849682257635943472981047, −6.91299499580354607991925018163, −5.94418705252991036971815092695, −5.33821814364066734230740631153, −3.86384224887747473863712941974, −2.98498826914679225386099669503, −2.18027440707632015444451926492, 0.092351155728314496677547824895, 1.88206805644213940190859256053, 2.88796662882581659221741232047, 4.34779032443784190412923204666, 4.91860981042033568685913350304, 5.84481499273409002196918090215, 7.03895067520234310642606245840, 7.77732685695488859118222980406, 8.678760336117358142509338435690, 9.271827671102272801144697178737

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