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Properties

Label 2-816-12.11-c1-0-15
Degree $2$
Conductor $816$
Sign $0.675 + 0.737i$
Analytic cond. $6.51579$
Root an. cond. $2.55260$
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  + (−1.65 − 0.521i)3-s − 0.575i·5-s + 0.301i·7-s + (2.45 + 1.72i)9-s − 1.39·11-s + 2.30·13-s + (−0.300 + 0.950i)15-s + i·17-s − 0.707i·19-s + (0.157 − 0.498i)21-s + 1.25·23-s + 4.66·25-s + (−3.15 − 4.12i)27-s − 4.73i·29-s − 0.394i·31-s + ⋯
L(s)  = 1  + (−0.953 − 0.301i)3-s − 0.257i·5-s + 0.114i·7-s + (0.818 + 0.574i)9-s − 0.419·11-s + 0.640·13-s + (−0.0774 + 0.245i)15-s + 0.242i·17-s − 0.162i·19-s + (0.0343 − 0.108i)21-s + 0.260·23-s + 0.933·25-s + (−0.607 − 0.794i)27-s − 0.878i·29-s − 0.0709i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $0.675 + 0.737i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ 0.675 + 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996006 - 0.438527i\)
\(L(\frac12)\) \(\approx\) \(0.996006 - 0.438527i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 0.521i)T \)
17 \( 1 - iT \)
good5 \( 1 + 0.575iT - 5T^{2} \)
7 \( 1 - 0.301iT - 7T^{2} \)
11 \( 1 + 1.39T + 11T^{2} \)
13 \( 1 - 2.30T + 13T^{2} \)
19 \( 1 + 0.707iT - 19T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 + 4.73iT - 29T^{2} \)
31 \( 1 + 0.394iT - 31T^{2} \)
37 \( 1 - 4.79T + 37T^{2} \)
41 \( 1 + 4.30iT - 41T^{2} \)
43 \( 1 + 8.76iT - 43T^{2} \)
47 \( 1 - 5.72T + 47T^{2} \)
53 \( 1 + 7.42iT - 53T^{2} \)
59 \( 1 - 4.06T + 59T^{2} \)
61 \( 1 + 3.71T + 61T^{2} \)
67 \( 1 + 7.90iT - 67T^{2} \)
71 \( 1 - 8.95T + 71T^{2} \)
73 \( 1 + 5.50T + 73T^{2} \)
79 \( 1 + 0.301iT - 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 0.377iT - 89T^{2} \)
97 \( 1 - 2.69T + 97T^{2} \)
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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

−10.39866193617496719543505465328, −9.309557584680044649109502636156, −8.377087938699637479896143254225, −7.45980022314414127937119485880, −6.56348696069322726355383581473, −5.71615439798074777927484309863, −4.92994456593408569215806401345, −3.85549787726431125315781426355, −2.25416573266696764987913721636, −0.77385560371768594374574041642, 1.10972678794916262552589706516, 2.91845233223388053695500418026, 4.12498149025626019020725260090, 5.05626795410542964273563962217, 5.95518912364477024985084528726, 6.77119919223454336600017415127, 7.62807068886723139702628770281, 8.782559149056244369914588590069, 9.654389518741454002506198783921, 10.59430488871054228373413600613

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