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Properties

Label 2-1160-145.12-c1-0-13
Degree $2$
Conductor $1160$
Sign $0.348 - 0.937i$
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  + 2.10·3-s + (−2.18 + 0.478i)5-s + (1.94 + 1.94i)7-s + 1.41·9-s + (0.883 + 0.883i)11-s + (2.69 + 2.69i)13-s + (−4.58 + 1.00i)15-s + 0.326i·17-s + (−5.28 + 5.28i)19-s + (4.09 + 4.09i)21-s + (−0.557 + 0.557i)23-s + (4.54 − 2.09i)25-s − 3.33·27-s + (−1.35 − 5.21i)29-s + (3.27 + 3.27i)31-s + ⋯
L(s)  = 1  + 1.21·3-s + (−0.976 + 0.214i)5-s + (0.736 + 0.736i)7-s + 0.471·9-s + (0.266 + 0.266i)11-s + (0.747 + 0.747i)13-s + (−1.18 + 0.259i)15-s + 0.0792i·17-s + (−1.21 + 1.21i)19-s + (0.893 + 0.893i)21-s + (−0.116 + 0.116i)23-s + (0.908 − 0.418i)25-s − 0.641·27-s + (−0.252 − 0.967i)29-s + (0.588 + 0.588i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1160\)    =    \(2^{3} \cdot 5 \cdot 29\)
Sign: $0.348 - 0.937i$
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: \((2,\ 1160,\ (\ :1/2),\ 0.348 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.110666508\)
\(L(\frac12)\) \(\approx\) \(2.110666508\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 0.478i)T \)
29 \( 1 + (1.35 + 5.21i)T \)
good3 \( 1 - 2.10T + 3T^{2} \)
7 \( 1 + (-1.94 - 1.94i)T + 7iT^{2} \)
11 \( 1 + (-0.883 - 0.883i)T + 11iT^{2} \)
13 \( 1 + (-2.69 - 2.69i)T + 13iT^{2} \)
17 \( 1 - 0.326iT - 17T^{2} \)
19 \( 1 + (5.28 - 5.28i)T - 19iT^{2} \)
23 \( 1 + (0.557 - 0.557i)T - 23iT^{2} \)
31 \( 1 + (-3.27 - 3.27i)T + 31iT^{2} \)
37 \( 1 - 6.69T + 37T^{2} \)
41 \( 1 + (6.35 - 6.35i)T - 41iT^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + (-1.79 + 1.79i)T - 53iT^{2} \)
59 \( 1 - 6.73iT - 59T^{2} \)
61 \( 1 + (-5.50 - 5.50i)T + 61iT^{2} \)
67 \( 1 + (-3.12 + 3.12i)T - 67iT^{2} \)
71 \( 1 - 3.62iT - 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + (-5.17 + 5.17i)T - 79iT^{2} \)
83 \( 1 + (-3.60 + 3.60i)T - 83iT^{2} \)
89 \( 1 + (-10.8 + 10.8i)T - 89iT^{2} \)
97 \( 1 + 3.89T + 97T^{2} \)
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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

−9.774811896197410877634230820611, −8.771400886931038899490712356891, −8.374529713219187162936773118439, −7.84446523544885539107334547150, −6.77582961032013623431286606667, −5.79002492895016118402652923669, −4.35440752662396868624274367060, −3.85516274642118343481263558149, −2.68894184871111146730505059729, −1.71866978040072504220136216780, 0.826707141795315107263985317847, 2.38132045155952489346617767103, 3.51872243692626643299599230570, 4.11818273547936216856812287205, 5.10256117341001122114583804376, 6.52131678105717021591850271786, 7.48340017791655134561201526292, 8.109448817202197421017328702972, 8.631381767314844192674889231725, 9.299463410468743271056410213659

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