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Properties

Label 2-1160-145.17-c1-0-42
Degree $2$
Conductor $1160$
Sign $-0.934 - 0.355i$
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  − 2i·3-s + (−1 − 2i)5-s + (−2.64 − 2.64i)7-s − 9-s + (2.64 − 2.64i)11-s + (−3 − 3i)13-s + (−4 + 2i)15-s + 5.29·17-s + (0.645 + 0.645i)19-s + (−5.29 + 5.29i)21-s + (2.64 − 2.64i)23-s + (−3 + 4i)25-s − 4i·27-s + (−5 + 2i)29-s + (−0.645 + 0.645i)31-s + ⋯
L(s)  = 1  − 1.15i·3-s + (−0.447 − 0.894i)5-s + (−0.999 − 0.999i)7-s − 0.333·9-s + (0.797 − 0.797i)11-s + (−0.832 − 0.832i)13-s + (−1.03 + 0.516i)15-s + 1.28·17-s + (0.148 + 0.148i)19-s + (−1.15 + 1.15i)21-s + (0.551 − 0.551i)23-s + (−0.600 + 0.800i)25-s − 0.769i·27-s + (−0.928 + 0.371i)29-s + (−0.115 + 0.115i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1160\)    =    \(2^{3} \cdot 5 \cdot 29\)
Sign: $-0.934 - 0.355i$
Analytic conductor: \(9.26264\)
Root analytic conductor: \(3.04345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1160} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1160,\ (\ :1/2),\ -0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.119168320\)
\(L(\frac12)\) \(\approx\) \(1.119168320\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
29 \( 1 + (5 - 2i)T \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 + (2.64 + 2.64i)T + 7iT^{2} \)
11 \( 1 + (-2.64 + 2.64i)T - 11iT^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 + (-0.645 - 0.645i)T + 19iT^{2} \)
23 \( 1 + (-2.64 + 2.64i)T - 23iT^{2} \)
31 \( 1 + (0.645 - 0.645i)T - 31iT^{2} \)
37 \( 1 - 7.29iT - 37T^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 8.58iT - 47T^{2} \)
53 \( 1 + (-4.29 + 4.29i)T - 53iT^{2} \)
59 \( 1 + 9.29iT - 59T^{2} \)
61 \( 1 + (10.2 - 10.2i)T - 61iT^{2} \)
67 \( 1 + (-6.64 + 6.64i)T - 67iT^{2} \)
71 \( 1 - 2.70iT - 71T^{2} \)
73 \( 1 - 5.29T + 73T^{2} \)
79 \( 1 + (-7.93 - 7.93i)T + 79iT^{2} \)
83 \( 1 + (0.645 - 0.645i)T - 83iT^{2} \)
89 \( 1 + (-12.2 - 12.2i)T + 89iT^{2} \)
97 \( 1 + 11.2iT - 97T^{2} \)
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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

−9.359387137698453111756267433634, −8.249149041269235697880627169444, −7.64993323139443162272023533465, −6.96794876073340127590017512918, −6.13570083671478047882417660675, −5.12585879117700172221556578267, −3.88856692606108410839498763203, −3.07322407990606287902249772719, −1.29225114951321863805244697210, −0.53625481985151742235443149275, 2.22800331230975428145997402751, 3.36403656067069270335554458027, 3.96274258849903090699211481377, 5.05841515871848422914887848681, 6.00678806927508193221369919914, 6.98326912102499908013925309844, 7.59456255993122673934570453485, 9.192586683627711871652532250526, 9.369269342711189210298898156827, 10.06732412443571459597463943581

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