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Properties

Label 2-5796-161.160-c1-0-57
Degree $2$
Conductor $5796$
Sign $0.998 - 0.0588i$
Analytic cond. $46.2812$
Root an. cond. $6.80303$
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  + 3.79·5-s + (2.36 + 1.18i)7-s + 1.64i·11-s − 4.11i·13-s + 2.96·17-s − 0.515·19-s + (−4.14 + 2.40i)23-s + 9.38·25-s + 2.86·29-s − 9.12i·31-s + (8.96 + 4.51i)35-s + 9.45i·37-s + 1.93i·41-s + 0.259i·43-s − 11.5i·47-s + ⋯
L(s)  = 1  + 1.69·5-s + (0.893 + 0.449i)7-s + 0.495i·11-s − 1.14i·13-s + 0.719·17-s − 0.118·19-s + (−0.865 + 0.501i)23-s + 1.87·25-s + 0.531·29-s − 1.63i·31-s + (1.51 + 0.762i)35-s + 1.55i·37-s + 0.302i·41-s + 0.0395i·43-s − 1.67i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5796\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 23\)
Sign: $0.998 - 0.0588i$
Analytic conductor: \(46.2812\)
Root analytic conductor: \(6.80303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5796} (5473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5796,\ (\ :1/2),\ 0.998 - 0.0588i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.394524912\)
\(L(\frac12)\) \(\approx\) \(3.394524912\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.36 - 1.18i)T \)
23 \( 1 + (4.14 - 2.40i)T \)
good5 \( 1 - 3.79T + 5T^{2} \)
11 \( 1 - 1.64iT - 11T^{2} \)
13 \( 1 + 4.11iT - 13T^{2} \)
17 \( 1 - 2.96T + 17T^{2} \)
19 \( 1 + 0.515T + 19T^{2} \)
29 \( 1 - 2.86T + 29T^{2} \)
31 \( 1 + 9.12iT - 31T^{2} \)
37 \( 1 - 9.45iT - 37T^{2} \)
41 \( 1 - 1.93iT - 41T^{2} \)
43 \( 1 - 0.259iT - 43T^{2} \)
47 \( 1 + 11.5iT - 47T^{2} \)
53 \( 1 - 2.20iT - 53T^{2} \)
59 \( 1 + 4.42iT - 59T^{2} \)
61 \( 1 - 6.63T + 61T^{2} \)
67 \( 1 - 9.71iT - 67T^{2} \)
71 \( 1 - 4.81T + 71T^{2} \)
73 \( 1 + 9.12iT - 73T^{2} \)
79 \( 1 + 12.3iT - 79T^{2} \)
83 \( 1 - 9.06T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 8.81T + 97T^{2} \)
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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

−8.071726769032907340466389486480, −7.57887516718836671018193672376, −6.43513277655055591490524966128, −5.93916169765842835598948268858, −5.26969669410857468417082679560, −4.81580001248539788167062169234, −3.57484239168909407908153893474, −2.50184094753964019067490628407, −1.97956260162926887352133357761, −1.03530630437548905734810207343, 1.06675556900393591274356196057, 1.82774490275146732411963844735, 2.52629096730312619032274148816, 3.69538557626372908376302251882, 4.60626304651138208546178334290, 5.28218170044310574678706043439, 5.95687229301718879180305989302, 6.59325591088863225158394028634, 7.29089788108031838673235021807, 8.234231733624105956265022139247

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