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Properties

Label 2-780-780.623-c0-0-1
Degree $2$
Conductor $780$
Sign $-0.584 - 0.811i$
Analytic cond. $0.389270$
Root an. cond. $0.623915$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 + 0.382i)2-s + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (−0.923 + 0.382i)5-s + (−0.923 + 0.382i)6-s + (0.382 + 0.923i)8-s − 1.00i·9-s − 10-s + 1.84i·11-s − 12-s + (−0.707 − 0.707i)13-s + (0.382 − 0.923i)15-s + i·16-s + (0.382 − 0.923i)18-s + (−0.923 − 0.382i)20-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)2-s + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (−0.923 + 0.382i)5-s + (−0.923 + 0.382i)6-s + (0.382 + 0.923i)8-s − 1.00i·9-s − 10-s + 1.84i·11-s − 12-s + (−0.707 − 0.707i)13-s + (0.382 − 0.923i)15-s + i·16-s + (0.382 − 0.923i)18-s + (−0.923 − 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(0.389270\)
Root analytic conductor: \(0.623915\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :0),\ -0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.084189923\)
\(L(\frac12)\) \(\approx\) \(1.084189923\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 - 0.382i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 - 1.84iT - T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.84T + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 0.765iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
89 \( 1 + 0.765iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.88192246542961899551236561688, −10.24299242530797312641836268850, −9.211740677811477021265448191962, −7.79035692331428616660805384950, −7.26570775918751243890094896669, −6.38473594573395854140513704524, −5.23561900098749589649437331303, −4.53233777382393395300249165279, −3.80816078315282093946308497366, −2.55981604785848211945927088360, 0.963637430751299600179650989659, 2.61197408844993564883939665468, 3.83575788377267315833126603121, 4.82226986219530633967222512238, 5.71134374192436262323547506838, 6.51580522601161658942486060423, 7.45254566894655162223462532979, 8.279448933893937888871606136105, 9.459426940639252288413509627634, 10.85191274929674357447953968193

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