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Properties

Label 2-1045-5.4-c1-0-79
Degree $2$
Conductor $1045$
Sign $-0.809 - 0.586i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
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  − 1.60i·2-s − 0.776i·3-s − 0.579·4-s + (−1.81 − 1.31i)5-s − 1.24·6-s + 0.953i·7-s − 2.28i·8-s + 2.39·9-s + (−2.10 + 2.90i)10-s − 11-s + 0.449i·12-s − 4.91i·13-s + 1.53·14-s + (−1.01 + 1.40i)15-s − 4.82·16-s − 1.30i·17-s + ⋯
L(s)  = 1  − 1.13i·2-s − 0.448i·3-s − 0.289·4-s + (−0.809 − 0.586i)5-s − 0.508·6-s + 0.360i·7-s − 0.806i·8-s + 0.799·9-s + (−0.666 + 0.919i)10-s − 0.301·11-s + 0.129i·12-s − 1.36i·13-s + 0.409·14-s + (−0.262 + 0.362i)15-s − 1.20·16-s − 0.316i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.809 - 0.586i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.809 - 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.183470309\)
\(L(\frac12)\) \(\approx\) \(1.183470309\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.81 + 1.31i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 + 1.60iT - 2T^{2} \)
3 \( 1 + 0.776iT - 3T^{2} \)
7 \( 1 - 0.953iT - 7T^{2} \)
13 \( 1 + 4.91iT - 13T^{2} \)
17 \( 1 + 1.30iT - 17T^{2} \)
23 \( 1 + 1.40iT - 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + 0.261T + 31T^{2} \)
37 \( 1 - 2.27iT - 37T^{2} \)
41 \( 1 + 3.22T + 41T^{2} \)
43 \( 1 - 1.39iT - 43T^{2} \)
47 \( 1 + 1.24iT - 47T^{2} \)
53 \( 1 - 9.04iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 + 3.94iT - 67T^{2} \)
71 \( 1 + 5.54T + 71T^{2} \)
73 \( 1 + 2.78iT - 73T^{2} \)
79 \( 1 + 7.73T + 79T^{2} \)
83 \( 1 + 9.84iT - 83T^{2} \)
89 \( 1 - 8.73T + 89T^{2} \)
97 \( 1 - 0.370iT - 97T^{2} \)
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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

−9.610891014056725388129078922376, −8.713900280854135194301116532894, −7.66958667440132219207868340955, −7.22779289434650168918567790166, −5.89486105692895364070425523851, −4.82251073143108540841447876600, −3.80895226192489210535499445007, −2.91580244515921904360089533625, −1.69075009471971268819137415051, −0.52804866672139824695521208590, 2.01745834634079632540345735515, 3.62548789812542421762185791016, 4.33542927232627080584532013290, 5.33679574879870534335318697611, 6.44798075029832358155680370568, 7.20450987476161231676246783349, 7.53737710427235849762100997893, 8.584413977047344691049718215413, 9.453974799892215628703402396106, 10.40364445616644021360492028951

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