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Properties

Label 2-780-65.8-c1-0-0
Degree $2$
Conductor $780$
Sign $-0.894 - 0.447i$
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 + (−0.705 + 2.12i)5-s + 3.56i·7-s + 1.00i·9-s + (2.90 − 2.90i)11-s + (−3.59 − 0.292i)13-s + (1.99 − 1.00i)15-s + (−2.74 − 2.74i)17-s + (−3.05 + 3.05i)19-s + (2.52 − 2.52i)21-s + (−3.58 + 3.58i)23-s + (−4.00 − 2.99i)25-s + (0.707 − 0.707i)27-s + 4.28i·29-s + (−6.44 − 6.44i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.315 + 0.948i)5-s + 1.34i·7-s + 0.333i·9-s + (0.877 − 0.877i)11-s + (−0.996 − 0.0811i)13-s + (0.516 − 0.258i)15-s + (−0.664 − 0.664i)17-s + (−0.700 + 0.700i)19-s + (0.550 − 0.550i)21-s + (−0.747 + 0.747i)23-s + (−0.801 − 0.598i)25-s + (0.136 − 0.136i)27-s + 0.796i·29-s + (−1.15 − 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 780,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116882 + 0.494516i\)
\(L(\frac12)\) \(\approx\) \(0.116882 + 0.494516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.705 - 2.12i)T \)
13 \( 1 + (3.59 + 0.292i)T \)
good7 \( 1 - 3.56iT - 7T^{2} \)
11 \( 1 + (-2.90 + 2.90i)T - 11iT^{2} \)
17 \( 1 + (2.74 + 2.74i)T + 17iT^{2} \)
19 \( 1 + (3.05 - 3.05i)T - 19iT^{2} \)
23 \( 1 + (3.58 - 3.58i)T - 23iT^{2} \)
29 \( 1 - 4.28iT - 29T^{2} \)
31 \( 1 + (6.44 + 6.44i)T + 31iT^{2} \)
37 \( 1 + 10.5iT - 37T^{2} \)
41 \( 1 + (-5.79 - 5.79i)T + 41iT^{2} \)
43 \( 1 + (7.39 - 7.39i)T - 43iT^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 + (-6.34 - 6.34i)T + 53iT^{2} \)
59 \( 1 + (3.62 + 3.62i)T + 59iT^{2} \)
61 \( 1 - 0.559T + 61T^{2} \)
67 \( 1 + 9.61T + 67T^{2} \)
71 \( 1 + (-0.724 - 0.724i)T + 71iT^{2} \)
73 \( 1 - 3.21T + 73T^{2} \)
79 \( 1 - 6.90iT - 79T^{2} \)
83 \( 1 + 4.46iT - 83T^{2} \)
89 \( 1 + (6.82 + 6.82i)T + 89iT^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−11.03575376653447912778890895301, −9.712241182472430764882578986227, −9.040799834873270147657523343112, −7.962443769925488054730158228334, −7.18876625851531480958419618439, −6.12821989660428669976234587963, −5.72594604088597293935205963467, −4.28202982795355034859782512368, −3.00518465694049232357461208750, −2.02842453922747100469205763749, 0.25636507455507952739324993426, 1.83992987659099892337661615664, 3.91442385908112107545679447665, 4.35115386069197288769152764316, 5.17412528328823738547288994137, 6.64165292184450564652001974583, 7.16481957338007134098671824639, 8.337411426793187006525270923252, 9.137457532429043709982418380128, 10.08409333343925758445279148048

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