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Properties

Label 2-780-12.11-c1-0-11
Degree $2$
Conductor $780$
Sign $0.981 - 0.191i$
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  + (0.508 − 1.31i)2-s + (−1.38 − 1.03i)3-s + (−1.48 − 1.34i)4-s − i·5-s + (−2.07 + 1.30i)6-s + 3.45i·7-s + (−2.52 + 1.27i)8-s + (0.846 + 2.87i)9-s + (−1.31 − 0.508i)10-s − 1.38·11-s + (0.662 + 3.40i)12-s + 13-s + (4.56 + 1.75i)14-s + (−1.03 + 1.38i)15-s + (0.395 + 3.98i)16-s + 1.21i·17-s + ⋯
L(s)  = 1  + (0.359 − 0.933i)2-s + (−0.800 − 0.599i)3-s + (−0.741 − 0.671i)4-s − 0.447i·5-s + (−0.846 + 0.531i)6-s + 1.30i·7-s + (−0.892 + 0.450i)8-s + (0.282 + 0.959i)9-s + (−0.417 − 0.160i)10-s − 0.417·11-s + (0.191 + 0.981i)12-s + 0.277·13-s + (1.22 + 0.470i)14-s + (−0.267 + 0.358i)15-s + (0.0988 + 0.995i)16-s + 0.293i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.981 - 0.191i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ 0.981 - 0.191i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782028 + 0.0755279i\)
\(L(\frac12)\) \(\approx\) \(0.782028 + 0.0755279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.508 + 1.31i)T \)
3 \( 1 + (1.38 + 1.03i)T \)
5 \( 1 + iT \)
13 \( 1 - T \)
good7 \( 1 - 3.45iT - 7T^{2} \)
11 \( 1 + 1.38T + 11T^{2} \)
17 \( 1 - 1.21iT - 17T^{2} \)
19 \( 1 - 6.79iT - 19T^{2} \)
23 \( 1 + 2.54T + 23T^{2} \)
29 \( 1 - 2.34iT - 29T^{2} \)
31 \( 1 - 3.03iT - 31T^{2} \)
37 \( 1 - 1.49T + 37T^{2} \)
41 \( 1 + 9.51iT - 41T^{2} \)
43 \( 1 - 4.80iT - 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 1.56iT - 53T^{2} \)
59 \( 1 - 4.10T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 + 0.680iT - 67T^{2} \)
71 \( 1 + 0.298T + 71T^{2} \)
73 \( 1 + 9.83T + 73T^{2} \)
79 \( 1 - 15.6iT - 79T^{2} \)
83 \( 1 + 0.226T + 83T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−10.48042838698146045214531274333, −9.729987943625788391307389225321, −8.622013061235794438391824025942, −8.025555053651034686064636449624, −6.45652260218284629365525359685, −5.61297842462726647160993598151, −5.17705210758845563939762487650, −3.86354857035947337521787200529, −2.41906197714663304723966206703, −1.45092366811990323761410993631, 0.40769667867546041842609522326, 3.15782328121344378877355170174, 4.19344214958577988827799217577, 4.83619431911081284074011707537, 5.92081907766294822230317872690, 6.77857887009343359897900955837, 7.34415686408093383280187933757, 8.406258949771179694400403682985, 9.593039781487614476037684766913, 10.16334735845571730669750172903

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