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Properties

Label 2-780-12.11-c1-0-43
Degree $2$
Conductor $780$
Sign $0.750 - 0.661i$
Analytic cond. $6.22833$
Root an. cond. $2.49566$
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  + (1.32 − 0.5i)2-s + 1.73i·3-s + (1.50 − 1.32i)4-s − i·5-s + (0.866 + 2.29i)6-s + 4.37i·7-s + (1.32 − 2.50i)8-s − 2.99·9-s + (−0.5 − 1.32i)10-s + 3.46·11-s + (2.29 + 2.59i)12-s + 13-s + (2.18 + 5.79i)14-s + 1.73·15-s + (0.500 − 3.96i)16-s + 1.58i·17-s + ⋯
L(s)  = 1  + (0.935 − 0.353i)2-s + 0.999i·3-s + (0.750 − 0.661i)4-s − 0.447i·5-s + (0.353 + 0.935i)6-s + 1.65i·7-s + (0.467 − 0.883i)8-s − 0.999·9-s + (−0.158 − 0.418i)10-s + 1.04·11-s + (0.661 + 0.749i)12-s + 0.277·13-s + (0.585 + 1.54i)14-s + 0.447·15-s + (0.125 − 0.992i)16-s + 0.383i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.750 - 0.661i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ 0.750 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60295 + 0.983824i\)
\(L(\frac12)\) \(\approx\) \(2.60295 + 0.983824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.5i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 + iT \)
13 \( 1 - T \)
good7 \( 1 - 4.37iT - 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 - 4.37iT - 19T^{2} \)
23 \( 1 - 0.913T + 23T^{2} \)
29 \( 1 - 5.58iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 9.16iT - 41T^{2} \)
43 \( 1 - 5.29iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 9.16T + 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 - 0.417T + 73T^{2} \)
79 \( 1 + 14.0iT - 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 4.41T + 97T^{2} \)
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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

−10.58022029561173118409271113028, −9.415794092550358694474614517517, −9.098122111119033657927814363582, −8.011606091643923470657908336581, −6.30820004598557062692200153925, −5.80216985514763196016229265879, −4.96235589395322468024884967459, −4.00279517505924733572360277240, −3.08676075258837603777303016244, −1.81801294859000014912338439784, 1.17613564049473480581774207999, 2.72383163683644340600145715345, 3.78001532910002469993627521923, 4.69265451722838924654089712544, 6.11145554977769969329658547627, 6.72740112056281355420594666699, 7.32061877947967997402741180626, 8.008998224984630837167700547240, 9.232602811709084767057081184663, 10.57595278096466688849027773487

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