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Properties

Label 2-780-12.11-c1-0-38
Degree $2$
Conductor $780$
Sign $0.912 + 0.408i$
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  + (−0.599 − 1.28i)2-s + (1.66 − 0.468i)3-s + (−1.28 + 1.53i)4-s + i·5-s + (−1.59 − 1.85i)6-s + 0.936i·7-s + (2.73 + 0.719i)8-s + (2.56 − 1.56i)9-s + (1.28 − 0.599i)10-s + 2.39·11-s + (−1.41 + 3.16i)12-s − 13-s + (1.19 − 0.561i)14-s + (0.468 + 1.66i)15-s + (−0.719 − 3.93i)16-s + 7.12i·17-s + ⋯
L(s)  = 1  + (−0.424 − 0.905i)2-s + (0.962 − 0.270i)3-s + (−0.640 + 0.768i)4-s + 0.447i·5-s + (−0.653 − 0.757i)6-s + 0.353i·7-s + (0.967 + 0.254i)8-s + (0.853 − 0.520i)9-s + (0.405 − 0.189i)10-s + 0.723·11-s + (−0.408 + 0.912i)12-s − 0.277·13-s + (0.320 − 0.150i)14-s + (0.120 + 0.430i)15-s + (−0.179 − 0.983i)16-s + 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.912 + 0.408i$
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: \((2,\ 780,\ (\ :1/2),\ 0.912 + 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66865 - 0.356772i\)
\(L(\frac12)\) \(\approx\) \(1.66865 - 0.356772i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.599 + 1.28i)T \)
3 \( 1 + (-1.66 + 0.468i)T \)
5 \( 1 - iT \)
13 \( 1 + T \)
good7 \( 1 - 0.936iT - 7T^{2} \)
11 \( 1 - 2.39T + 11T^{2} \)
17 \( 1 - 7.12iT - 17T^{2} \)
19 \( 1 - 2.39iT - 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 + 4.27iT - 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + 12.4iT - 43T^{2} \)
47 \( 1 - 9.47T + 47T^{2} \)
53 \( 1 - 3.12iT - 53T^{2} \)
59 \( 1 + 0.525T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 9.47iT - 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 + 5.12iT - 89T^{2} \)
97 \( 1 - 0.246T + 97T^{2} \)
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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

−10.30277130056002607620374573788, −9.246705455256192325968192702840, −8.797106503367630487508109660836, −7.88010469501827463135488122100, −7.11258757673462687420939068574, −5.91516466496880962190418901578, −4.24011822393358727714830156110, −3.55623395127681574156278991055, −2.46101233727881499922331160494, −1.47304210336311113199632678414, 1.05162577665891452580837905126, 2.74374630475613348289149669579, 4.29558108286860200805128059207, 4.78341544819027236341226427320, 6.12438139520698931693669798616, 7.22704277888856324271041342511, 7.70774685713706183010175859798, 8.732377882765724756429318312134, 9.400773677266511269118361409908, 9.804989017310688466793445811883

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