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Properties

Label 2-780-65.8-c1-0-7
Degree $2$
Conductor $780$
Sign $0.465 + 0.885i$
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 + (−2.16 + 0.544i)5-s + 0.657i·7-s + 1.00i·9-s + (−0.997 + 0.997i)11-s + (3.44 − 1.06i)13-s + (1.91 + 1.14i)15-s + (−1.65 − 1.65i)17-s + (3.13 − 3.13i)19-s + (0.465 − 0.465i)21-s + (4.15 − 4.15i)23-s + (4.40 − 2.36i)25-s + (0.707 − 0.707i)27-s + 1.15i·29-s + (−3.59 − 3.59i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.969 + 0.243i)5-s + 0.248i·7-s + 0.333i·9-s + (−0.300 + 0.300i)11-s + (0.955 − 0.295i)13-s + (0.495 + 0.296i)15-s + (−0.400 − 0.400i)17-s + (0.719 − 0.719i)19-s + (0.101 − 0.101i)21-s + (0.866 − 0.866i)23-s + (0.881 − 0.472i)25-s + (0.136 − 0.136i)27-s + 0.214i·29-s + (−0.645 − 0.645i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.465 + 0.885i$
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: \((2,\ 780,\ (\ :1/2),\ 0.465 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851767 - 0.514614i\)
\(L(\frac12)\) \(\approx\) \(0.851767 - 0.514614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 - 0.544i)T \)
13 \( 1 + (-3.44 + 1.06i)T \)
good7 \( 1 - 0.657iT - 7T^{2} \)
11 \( 1 + (0.997 - 0.997i)T - 11iT^{2} \)
17 \( 1 + (1.65 + 1.65i)T + 17iT^{2} \)
19 \( 1 + (-3.13 + 3.13i)T - 19iT^{2} \)
23 \( 1 + (-4.15 + 4.15i)T - 23iT^{2} \)
29 \( 1 - 1.15iT - 29T^{2} \)
31 \( 1 + (3.59 + 3.59i)T + 31iT^{2} \)
37 \( 1 + 1.79iT - 37T^{2} \)
41 \( 1 + (3.13 + 3.13i)T + 41iT^{2} \)
43 \( 1 + (-3.88 + 3.88i)T - 43iT^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (-5.75 - 5.75i)T + 53iT^{2} \)
59 \( 1 + (-1.68 - 1.68i)T + 59iT^{2} \)
61 \( 1 + 4.76T + 61T^{2} \)
67 \( 1 + 1.08T + 67T^{2} \)
71 \( 1 + (2.40 + 2.40i)T + 71iT^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 + 3.38iT - 79T^{2} \)
83 \( 1 - 6.44iT - 83T^{2} \)
89 \( 1 + (5.50 + 5.50i)T + 89iT^{2} \)
97 \( 1 + 1.48T + 97T^{2} \)
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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

−10.47764062098067184208669084520, −9.103508184776560851287567839792, −8.438908685525672390235632924327, −7.37171985325665746608857502478, −6.90092230952417306338593852509, −5.71861923548822618780496383346, −4.78165574810860756506380066223, −3.64789492799828157759610268064, −2.47122603927006168895267519871, −0.64295732572575305465243034764, 1.17436120835648781833164002578, 3.26064992967703953320747086637, 3.99597197840969699607774672146, 5.00832511447938088860814840542, 5.96513903640677878256353987583, 7.03165298697295981558797791658, 7.922406072798958206355516797045, 8.734535679434271341597777952806, 9.575984337748807847787682686441, 10.71210388631946800047426977894

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