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Properties

Label 2-783-9.7-c1-0-12
Degree $2$
Conductor $783$
Sign $0.917 - 0.397i$
Analytic cond. $6.25228$
Root an. cond. $2.50045$
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  + (0.540 − 0.935i)2-s + (0.416 + 0.721i)4-s + (1.91 + 3.32i)5-s + (1.07 − 1.86i)7-s + 3.06·8-s + 4.14·10-s + (−1.42 + 2.47i)11-s + (0.862 + 1.49i)13-s + (−1.16 − 2.01i)14-s + (0.819 − 1.41i)16-s − 1.10·17-s − 2.98·19-s + (−1.59 + 2.76i)20-s + (1.54 + 2.67i)22-s + (−4.05 − 7.02i)23-s + ⋯
L(s)  = 1  + (0.381 − 0.661i)2-s + (0.208 + 0.360i)4-s + (0.857 + 1.48i)5-s + (0.407 − 0.706i)7-s + 1.08·8-s + 1.31·10-s + (−0.430 + 0.746i)11-s + (0.239 + 0.414i)13-s + (−0.311 − 0.539i)14-s + (0.204 − 0.354i)16-s − 0.268·17-s − 0.685·19-s + (−0.357 + 0.619i)20-s + (0.329 + 0.570i)22-s + (−0.846 − 1.46i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(6.25228\)
Root analytic conductor: \(2.50045\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 783,\ (\ :1/2),\ 0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40387 + 0.498252i\)
\(L(\frac12)\) \(\approx\) \(2.40387 + 0.498252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.540 + 0.935i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.91 - 3.32i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.07 + 1.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.42 - 2.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.862 - 1.49i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.10T + 17T^{2} \)
19 \( 1 + 2.98T + 19T^{2} \)
23 \( 1 + (4.05 + 7.02i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (-1.57 - 2.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.10T + 37T^{2} \)
41 \( 1 + (4.70 + 8.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.66 + 6.34i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.381 + 0.660i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (0.153 + 0.265i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.35 - 5.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.53 + 9.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + 3.16T + 73T^{2} \)
79 \( 1 + (-1.18 + 2.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.09 + 8.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 + (-8.44 + 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.52379519277933195321293260299, −10.05308295893779081406395169629, −8.653998372243683284492073783600, −7.48256751108939244471141594041, −6.96252203357071460185890695642, −6.09700841224644501373370385697, −4.62882366543353380893150139998, −3.79381134065492246834428010769, −2.55816544281774755915685673352, −1.97192065404092792583003535310, 1.20584333537798005777653225937, 2.30938484622282165646138518911, 4.23797115957560636723483233097, 5.21830325966782384832542987942, 5.69476404474304002344430531991, 6.32118131813416578946260230562, 7.86213496849249204896183772206, 8.376272574769494435945416959233, 9.348731757854691982907989846461, 10.05863621754785830512200319746

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