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Properties

Label 2-780-12.11-c1-0-3
Degree $2$
Conductor $780$
Sign $-0.960 - 0.277i$
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  + (−0.177 + 1.40i)2-s + (0.0510 − 1.73i)3-s + (−1.93 − 0.499i)4-s − i·5-s + (2.41 + 0.379i)6-s + 4.12i·7-s + (1.04 − 2.62i)8-s + (−2.99 − 0.176i)9-s + (1.40 + 0.177i)10-s − 1.68·11-s + (−0.962 + 3.32i)12-s − 13-s + (−5.78 − 0.733i)14-s + (−1.73 − 0.0510i)15-s + (3.50 + 1.93i)16-s − 2.57i·17-s + ⋯
L(s)  = 1  + (−0.125 + 0.992i)2-s + (0.0294 − 0.999i)3-s + (−0.968 − 0.249i)4-s − 0.447i·5-s + (0.987 + 0.154i)6-s + 1.55i·7-s + (0.369 − 0.929i)8-s + (−0.998 − 0.0588i)9-s + (0.443 + 0.0562i)10-s − 0.507·11-s + (−0.277 + 0.960i)12-s − 0.277·13-s + (−1.54 − 0.196i)14-s + (−0.447 − 0.0131i)15-s + (0.875 + 0.483i)16-s − 0.623i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.960 - 0.277i$
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: \((2,\ 780,\ (\ :1/2),\ -0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0617850 + 0.435881i\)
\(L(\frac12)\) \(\approx\) \(0.0617850 + 0.435881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.177 - 1.40i)T \)
3 \( 1 + (-0.0510 + 1.73i)T \)
5 \( 1 + iT \)
13 \( 1 + T \)
good7 \( 1 - 4.12iT - 7T^{2} \)
11 \( 1 + 1.68T + 11T^{2} \)
17 \( 1 + 2.57iT - 17T^{2} \)
19 \( 1 - 1.95iT - 19T^{2} \)
23 \( 1 + 6.44T + 23T^{2} \)
29 \( 1 - 5.70iT - 29T^{2} \)
31 \( 1 - 8.50iT - 31T^{2} \)
37 \( 1 + 4.19T + 37T^{2} \)
41 \( 1 + 8.18iT - 41T^{2} \)
43 \( 1 - 7.55iT - 43T^{2} \)
47 \( 1 + 7.23T + 47T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + 0.569T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 - 3.82iT - 67T^{2} \)
71 \( 1 - 6.38T + 71T^{2} \)
73 \( 1 - 4.41T + 73T^{2} \)
79 \( 1 + 3.60iT - 79T^{2} \)
83 \( 1 - 4.05T + 83T^{2} \)
89 \( 1 + 18.1iT - 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−10.55041529867353496753309027561, −9.367038325885222868246354379746, −8.755588030018829785118438485637, −8.110274468080667032603661623485, −7.29927209544204949104669687271, −6.27484388282609503043117595471, −5.57772992519594116285245871956, −4.89147711739303839040327271715, −3.09766591367164330908547056411, −1.71918122806909954306082454356, 0.22339685956145705342667318968, 2.23582303330831837700762166571, 3.52816530402995424100685882869, 4.10250473470743530252540783365, 4.99558605322767427163494833298, 6.27391759115236759592363482535, 7.70838652767342086495400112123, 8.243812853682451887529082704336, 9.642356479955066049132085031683, 9.966165880171784735782752763941

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