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Properties

Label 2-780-65.8-c1-0-6
Degree $2$
Conductor $780$
Sign $0.967 - 0.251i$
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 + (−2.04 + 0.898i)5-s − 3.12i·7-s + 1.00i·9-s + (2.81 − 2.81i)11-s + (−0.350 + 3.58i)13-s + (−2.08 − 0.812i)15-s + (5.74 + 5.74i)17-s + (3.34 − 3.34i)19-s + (2.20 − 2.20i)21-s + (1.98 − 1.98i)23-s + (3.38 − 3.68i)25-s + (−0.707 + 0.707i)27-s + 3.27i·29-s + (2.64 + 2.64i)31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.915 + 0.401i)5-s − 1.17i·7-s + 0.333i·9-s + (0.849 − 0.849i)11-s + (−0.0972 + 0.995i)13-s + (−0.537 − 0.209i)15-s + (1.39 + 1.39i)17-s + (0.767 − 0.767i)19-s + (0.481 − 0.481i)21-s + (0.413 − 0.413i)23-s + (0.676 − 0.736i)25-s + (−0.136 + 0.136i)27-s + 0.607i·29-s + (0.475 + 0.475i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.967 - 0.251i$
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: \((2,\ 780,\ (\ :1/2),\ 0.967 - 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62350 + 0.207336i\)
\(L(\frac12)\) \(\approx\) \(1.62350 + 0.207336i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 - 0.898i)T \)
13 \( 1 + (0.350 - 3.58i)T \)
good7 \( 1 + 3.12iT - 7T^{2} \)
11 \( 1 + (-2.81 + 2.81i)T - 11iT^{2} \)
17 \( 1 + (-5.74 - 5.74i)T + 17iT^{2} \)
19 \( 1 + (-3.34 + 3.34i)T - 19iT^{2} \)
23 \( 1 + (-1.98 + 1.98i)T - 23iT^{2} \)
29 \( 1 - 3.27iT - 29T^{2} \)
31 \( 1 + (-2.64 - 2.64i)T + 31iT^{2} \)
37 \( 1 + 9.27iT - 37T^{2} \)
41 \( 1 + (-2.25 - 2.25i)T + 41iT^{2} \)
43 \( 1 + (-1.66 + 1.66i)T - 43iT^{2} \)
47 \( 1 - 8.71iT - 47T^{2} \)
53 \( 1 + (-6.71 - 6.71i)T + 53iT^{2} \)
59 \( 1 + (2.43 + 2.43i)T + 59iT^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 4.99T + 67T^{2} \)
71 \( 1 + (2.49 + 2.49i)T + 71iT^{2} \)
73 \( 1 - 1.25T + 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + 4.80iT - 83T^{2} \)
89 \( 1 + (-2.08 - 2.08i)T + 89iT^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−10.58012117643628454621687989003, −9.420501320544408455572465187108, −8.657515341480071083083340189375, −7.68640415316437342663075973116, −7.08067812445964289462901221097, −6.05182206946879592978074262284, −4.54400688748857220504454329772, −3.83818251657611659936143945593, −3.13143826458151125516128759031, −1.12396377065064166923982283940, 1.11792754755705768501659246465, 2.71333695711162456988049367923, 3.61675555987974103478204764455, 4.95728839714004758596889443507, 5.72521908219919407589043625014, 7.05639171298134664276820927744, 7.76369821851727434751653810515, 8.432804729729673657210897006679, 9.423089918727572503605891445804, 9.935611204842462693985073572179

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