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Properties

Label 2-786-393.392-c1-0-41
Degree $2$
Conductor $786$
Sign $-0.145 + 0.989i$
Analytic cond. $6.27624$
Root an. cond. $2.50524$
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-s + (0.622 − 1.61i)3-s + 4-s − 3.59i·5-s + (0.622 − 1.61i)6-s + 1.80·7-s + 8-s + (−2.22 − 2.01i)9-s − 3.59i·10-s + 1.54i·11-s + (0.622 − 1.61i)12-s + 6.27·13-s + 1.80·14-s + (−5.80 − 2.23i)15-s + 16-s − 3.71·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.359 − 0.933i)3-s + 0.5·4-s − 1.60i·5-s + (0.254 − 0.659i)6-s + 0.682·7-s + 0.353·8-s + (−0.741 − 0.670i)9-s − 1.13i·10-s + 0.466i·11-s + (0.179 − 0.466i)12-s + 1.73·13-s + 0.482·14-s + (−1.49 − 0.577i)15-s + 0.250·16-s − 0.900·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(786\)    =    \(2 \cdot 3 \cdot 131\)
Sign: $-0.145 + 0.989i$
Analytic conductor: \(6.27624\)
Root analytic conductor: \(2.50524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{786} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 786,\ (\ :1/2),\ -0.145 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84718 - 2.13806i\)
\(L(\frac12)\) \(\approx\) \(1.84718 - 2.13806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.622 + 1.61i)T \)
131 \( 1 + (11.1 - 2.51i)T \)
good5 \( 1 + 3.59iT - 5T^{2} \)
7 \( 1 - 1.80T + 7T^{2} \)
11 \( 1 - 1.54iT - 11T^{2} \)
13 \( 1 - 6.27T + 13T^{2} \)
17 \( 1 + 3.71T + 17T^{2} \)
19 \( 1 - 7.82iT - 19T^{2} \)
23 \( 1 + 8.49T + 23T^{2} \)
29 \( 1 - 8.93T + 29T^{2} \)
31 \( 1 + 1.71iT - 31T^{2} \)
37 \( 1 + 7.39iT - 37T^{2} \)
41 \( 1 - 4.27iT - 41T^{2} \)
43 \( 1 + 4.42T + 43T^{2} \)
47 \( 1 - 5.35T + 47T^{2} \)
53 \( 1 - 7.16iT - 53T^{2} \)
59 \( 1 - 6.34iT - 59T^{2} \)
61 \( 1 - 2.24T + 61T^{2} \)
67 \( 1 - 3.75iT - 67T^{2} \)
71 \( 1 - 5.76T + 71T^{2} \)
73 \( 1 - 0.165iT - 73T^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + 0.870T + 83T^{2} \)
89 \( 1 - 1.13iT - 89T^{2} \)
97 \( 1 + 5.01iT - 97T^{2} \)
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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

−10.05643425654242110632142986915, −8.774754834223584049091389685897, −8.338147939614800396063337229007, −7.67651881280126766197430998812, −6.23894596832168453666720113836, −5.77106541424038337125342350588, −4.50973004952666536826550847300, −3.79200152505107490716972350633, −1.99982509646224773194086195929, −1.24180679068751095415537589529, 2.27533279269706102174151637952, 3.19263226346586785095638196541, 3.99421658868509907093570876198, 4.98517782964869297219908991823, 6.22117073775955574310794147749, 6.72372953064645889462399585710, 8.087570781205890040881534890544, 8.678714620581084569954103527220, 9.994567852558199152685684887732, 10.75487482180389068212514187001

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