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Properties

Label 2-780-65.8-c1-0-2
Degree $2$
Conductor $780$
Sign $0.0840 - 0.996i$
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  + (−0.707 − 0.707i)3-s + (1.17 − 1.90i)5-s + 4.16i·7-s + 1.00i·9-s + (−3.15 + 3.15i)11-s + (1.82 + 3.10i)13-s + (−2.17 + 0.513i)15-s + (−4.18 − 4.18i)17-s + (−4.37 + 4.37i)19-s + (2.94 − 2.94i)21-s + (−1.31 + 1.31i)23-s + (−2.23 − 4.47i)25-s + (0.707 − 0.707i)27-s + 3.93i·29-s + (3.84 + 3.84i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.525 − 0.850i)5-s + 1.57i·7-s + 0.333i·9-s + (−0.950 + 0.950i)11-s + (0.507 + 0.861i)13-s + (−0.561 + 0.132i)15-s + (−1.01 − 1.01i)17-s + (−1.00 + 1.00i)19-s + (0.642 − 0.642i)21-s + (−0.273 + 0.273i)23-s + (−0.446 − 0.894i)25-s + (0.136 − 0.136i)27-s + 0.730i·29-s + (0.691 + 0.691i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0840 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0840 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.0840 - 0.996i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ 0.0840 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.716107 + 0.658253i\)
\(L(\frac12)\) \(\approx\) \(0.716107 + 0.658253i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.17 + 1.90i)T \)
13 \( 1 + (-1.82 - 3.10i)T \)
good7 \( 1 - 4.16iT - 7T^{2} \)
11 \( 1 + (3.15 - 3.15i)T - 11iT^{2} \)
17 \( 1 + (4.18 + 4.18i)T + 17iT^{2} \)
19 \( 1 + (4.37 - 4.37i)T - 19iT^{2} \)
23 \( 1 + (1.31 - 1.31i)T - 23iT^{2} \)
29 \( 1 - 3.93iT - 29T^{2} \)
31 \( 1 + (-3.84 - 3.84i)T + 31iT^{2} \)
37 \( 1 - 8.44iT - 37T^{2} \)
41 \( 1 + (-1.55 - 1.55i)T + 41iT^{2} \)
43 \( 1 + (-8.11 + 8.11i)T - 43iT^{2} \)
47 \( 1 - 1.02iT - 47T^{2} \)
53 \( 1 + (1.36 + 1.36i)T + 53iT^{2} \)
59 \( 1 + (1.43 + 1.43i)T + 59iT^{2} \)
61 \( 1 - 0.293T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 + (-11.3 - 11.3i)T + 71iT^{2} \)
73 \( 1 - 6.46T + 73T^{2} \)
79 \( 1 - 8.32iT - 79T^{2} \)
83 \( 1 + 15.1iT - 83T^{2} \)
89 \( 1 + (-6.44 - 6.44i)T + 89iT^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−10.45770272410091592609076391091, −9.512191238533591422967717479250, −8.800611706395359263680016338900, −8.146888432604499964418661853633, −6.85279700687243769131074954319, −6.02618178571237138680825313327, −5.20705373572118873263473195045, −4.49038373115501187443798740722, −2.50626552296956607838992443210, −1.76394969480117908109372563687, 0.49220538557028014139848641404, 2.48714752506420234114178527474, 3.68499914079758603526537307547, 4.52652207381577142589928528949, 5.91598010358276237359796402728, 6.39066444215662879162037414435, 7.48715994518630666972155969685, 8.291175684736048723824663584604, 9.474042727042574324184670329923, 10.59569698939439689710977474616

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