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Properties

Label 2-770-11.5-c1-0-21
Degree $2$
Conductor $770$
Sign $-0.836 - 0.548i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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  + (−0.309 − 0.951i)2-s + (−0.5 − 0.363i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.190 + 0.587i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 2.48i)9-s + 0.999·10-s + (2.19 + 2.48i)11-s + 0.618·12-s + (−1.61 − 4.97i)13-s + (0.809 + 0.587i)14-s + (0.5 − 0.363i)15-s + (0.309 − 0.951i)16-s + (−0.5 + 1.53i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.288 − 0.209i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (−0.0779 + 0.239i)6-s + (−0.305 + 0.222i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.829i)9-s + 0.316·10-s + (0.660 + 0.750i)11-s + 0.178·12-s + (−0.448 − 1.38i)13-s + (0.216 + 0.157i)14-s + (0.129 − 0.0937i)15-s + (0.0772 − 0.237i)16-s + (−0.121 + 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.836 - 0.548i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0395976 + 0.132507i\)
\(L(\frac12)\) \(\approx\) \(0.0395976 + 0.132507i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-2.19 - 2.48i)T \)
good3 \( 1 + (0.5 + 0.363i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (1.61 + 4.97i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.5 - 1.53i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.309 - 0.224i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 8.47T + 23T^{2} \)
29 \( 1 + (3.23 - 2.35i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.38 + 4.25i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.85 - 5.70i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.97 + 5.06i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 + (-5.47 - 3.97i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2 + 6.15i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.92 + 2.12i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.47 - 7.60i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.09T + 67T^{2} \)
71 \( 1 + (-0.763 + 2.35i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.5 - 2.54i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.527 - 1.62i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.42 + 10.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 9.85T + 89T^{2} \)
97 \( 1 + (3.33 + 10.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.00711595044353647265070021315, −9.141390075859026593202239837390, −8.179863526692453802786019408695, −7.23679214184163141535298970761, −6.29927317452500345128251371497, −5.37681471450471154822027279526, −3.96663028541244997337203786977, −3.16883552945888590116645985384, −1.83519326927910516605314212812, −0.07499409271853391688244565101, 1.88541572452671039132428535357, 3.73852437391705222709004355582, 4.63666605358893546052627916550, 5.57635968605080073820213557594, 6.46988549593738776838516397097, 7.34170823022748628121132723929, 8.300389006033699837188105119767, 9.041667701780342938643960879210, 9.811827655061618183104177594845, 10.72298160152140929324525348312

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