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Properties

Label 2-780-65.8-c1-0-11
Degree $2$
Conductor $780$
Sign $0.862 + 0.506i$
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 + (1.95 + 1.09i)5-s − 4.81i·7-s + 1.00i·9-s + (1.53 − 1.53i)11-s + (−0.151 − 3.60i)13-s + (0.606 + 2.15i)15-s + (−3.32 − 3.32i)17-s + (4.77 − 4.77i)19-s + (3.40 − 3.40i)21-s + (−3.58 + 3.58i)23-s + (2.60 + 4.26i)25-s + (−0.707 + 0.707i)27-s + 3.23i·29-s + (−2.30 − 2.30i)31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.872 + 0.488i)5-s − 1.81i·7-s + 0.333i·9-s + (0.462 − 0.462i)11-s + (−0.0420 − 0.999i)13-s + (0.156 + 0.555i)15-s + (−0.805 − 0.805i)17-s + (1.09 − 1.09i)19-s + (0.742 − 0.742i)21-s + (−0.747 + 0.747i)23-s + (0.521 + 0.853i)25-s + (−0.136 + 0.136i)27-s + 0.599i·29-s + (−0.414 − 0.414i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.862 + 0.506i$
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: \((2,\ 780,\ (\ :1/2),\ 0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95277 - 0.531052i\)
\(L(\frac12)\) \(\approx\) \(1.95277 - 0.531052i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 - 1.09i)T \)
13 \( 1 + (0.151 + 3.60i)T \)
good7 \( 1 + 4.81iT - 7T^{2} \)
11 \( 1 + (-1.53 + 1.53i)T - 11iT^{2} \)
17 \( 1 + (3.32 + 3.32i)T + 17iT^{2} \)
19 \( 1 + (-4.77 + 4.77i)T - 19iT^{2} \)
23 \( 1 + (3.58 - 3.58i)T - 23iT^{2} \)
29 \( 1 - 3.23iT - 29T^{2} \)
31 \( 1 + (2.30 + 2.30i)T + 31iT^{2} \)
37 \( 1 - 6.36iT - 37T^{2} \)
41 \( 1 + (-2.79 - 2.79i)T + 41iT^{2} \)
43 \( 1 + (0.728 - 0.728i)T - 43iT^{2} \)
47 \( 1 + 1.10iT - 47T^{2} \)
53 \( 1 + (-7.40 - 7.40i)T + 53iT^{2} \)
59 \( 1 + (-9.82 - 9.82i)T + 59iT^{2} \)
61 \( 1 + 0.306T + 61T^{2} \)
67 \( 1 - 1.00T + 67T^{2} \)
71 \( 1 + (-9.17 - 9.17i)T + 71iT^{2} \)
73 \( 1 + 7.22T + 73T^{2} \)
79 \( 1 - 13.9iT - 79T^{2} \)
83 \( 1 + 16.2iT - 83T^{2} \)
89 \( 1 + (4.25 + 4.25i)T + 89iT^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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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

−10.06969363770017768363660789100, −9.668085995409543646542168096092, −8.636510305796093228807984437121, −7.39501071873260937601591254452, −7.01983844555511364964582690167, −5.77627586839335227351271104514, −4.73058468957940180263689798781, −3.64520351271384253364818878065, −2.76802522743145376042587022012, −1.04919398778249584150842594574, 1.84212214183905409565720040642, 2.28279420145391133702703153441, 3.91103098913983913915986668555, 5.19969348160787164523292128125, 6.01095589242991968622728425195, 6.67854064051459560002780529998, 8.058421750885413991610717413194, 8.828815720626197902416434260822, 9.305865018752600910550949868794, 10.05862967781619505974440177305

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