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Properties

Label 2-780-12.11-c1-0-45
Degree $2$
Conductor $780$
Sign $0.305 + 0.952i$
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  + (−1.36 + 0.367i)2-s + (−1.69 − 0.370i)3-s + (1.72 − 1.00i)4-s − i·5-s + (2.44 − 0.116i)6-s − 2.50i·7-s + (−1.99 + 2.00i)8-s + (2.72 + 1.25i)9-s + (0.367 + 1.36i)10-s + 3.25·11-s + (−3.29 + 1.05i)12-s + 13-s + (0.922 + 3.42i)14-s + (−0.370 + 1.69i)15-s + (1.98 − 3.47i)16-s − 2.15i·17-s + ⋯
L(s)  = 1  + (−0.965 + 0.260i)2-s + (−0.976 − 0.213i)3-s + (0.864 − 0.502i)4-s − 0.447i·5-s + (0.998 − 0.0475i)6-s − 0.948i·7-s + (−0.704 + 0.709i)8-s + (0.908 + 0.417i)9-s + (0.116 + 0.431i)10-s + 0.979·11-s + (−0.952 + 0.305i)12-s + 0.277·13-s + (0.246 + 0.915i)14-s + (−0.0956 + 0.436i)15-s + (0.495 − 0.868i)16-s − 0.523i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.305 + 0.952i$
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: \((2,\ 780,\ (\ :1/2),\ 0.305 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.582272 - 0.424598i\)
\(L(\frac12)\) \(\approx\) \(0.582272 - 0.424598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.367i)T \)
3 \( 1 + (1.69 + 0.370i)T \)
5 \( 1 + iT \)
13 \( 1 - T \)
good7 \( 1 + 2.50iT - 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
17 \( 1 + 2.15iT - 17T^{2} \)
19 \( 1 - 4.69iT - 19T^{2} \)
23 \( 1 - 5.22T + 23T^{2} \)
29 \( 1 - 0.00443iT - 29T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 - 0.0889T + 37T^{2} \)
41 \( 1 + 9.34iT - 41T^{2} \)
43 \( 1 + 2.07iT - 43T^{2} \)
47 \( 1 + 6.13T + 47T^{2} \)
53 \( 1 + 6.54iT - 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 1.16T + 61T^{2} \)
67 \( 1 + 0.0288iT - 67T^{2} \)
71 \( 1 + 7.72T + 71T^{2} \)
73 \( 1 - 7.70T + 73T^{2} \)
79 \( 1 + 14.8iT - 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + 17.5iT - 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−10.17602637698719277623675045532, −9.354577146875071366599250924520, −8.423850387286458776516782733030, −7.36787458548382455798068899312, −6.84228889994842139899171219271, −5.92899815638899541249955207275, −4.96142385924413451005232829765, −3.72549160340577633531831553181, −1.66797928838145087076228910208, −0.68319414284163517562291158245, 1.21991629249028814642786175611, 2.68413041522038137360903694288, 3.94300112100383149278749483118, 5.31047270245842970498543286950, 6.43576213791055718366498311110, 6.76295664410223010705097256841, 8.010312849730408989255901960209, 9.093258769583286158628370809330, 9.518701187995290015665358073777, 10.58614067594969554381239567940

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