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Properties

Label 2-819-91.25-c1-0-24
Degree $2$
Conductor $819$
Sign $0.966 - 0.256i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
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  + (−1.84 + 1.06i)2-s + (1.25 − 2.18i)4-s + (3.12 − 1.80i)5-s + (1.20 − 2.35i)7-s + 1.10i·8-s + (−3.83 + 6.63i)10-s + (3.45 + 1.99i)11-s + (−2.51 + 2.58i)13-s + (0.274 + 5.61i)14-s + (1.34 + 2.33i)16-s + (−2.39 + 4.14i)17-s + (2.72 − 1.57i)19-s − 9.07i·20-s − 8.48·22-s + (1.08 + 1.88i)23-s + ⋯
L(s)  = 1  + (−1.30 + 0.751i)2-s + (0.629 − 1.09i)4-s + (1.39 − 0.806i)5-s + (0.457 − 0.889i)7-s + 0.389i·8-s + (−1.21 + 2.09i)10-s + (1.04 + 0.601i)11-s + (−0.698 + 0.715i)13-s + (0.0734 + 1.50i)14-s + (0.337 + 0.583i)16-s + (−0.580 + 1.00i)17-s + (0.625 − 0.361i)19-s − 2.03i·20-s − 1.80·22-s + (0.227 + 0.393i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11982 + 0.146240i\)
\(L(\frac12)\) \(\approx\) \(1.11982 + 0.146240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.20 + 2.35i)T \)
13 \( 1 + (2.51 - 2.58i)T \)
good2 \( 1 + (1.84 - 1.06i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-3.12 + 1.80i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.45 - 1.99i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.39 - 4.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.72 + 1.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.08 - 1.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.57T + 29T^{2} \)
31 \( 1 + (-1.28 - 0.743i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.29 + 2.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.11iT - 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 + (0.882 - 0.509i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.01 + 5.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.24 + 2.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.01 - 1.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.38 - 1.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.80iT - 71T^{2} \)
73 \( 1 + (2.67 + 1.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.984 + 1.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.66iT - 83T^{2} \)
89 \( 1 + (-11.0 + 6.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.35iT - 97T^{2} \)
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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

−9.913363707439251430014647073881, −9.331410958635185047836372322999, −8.773737541952864984170355808785, −7.79324561138431133092470167918, −6.83064278387066036037007910685, −6.33772493598989646115097598087, −5.08418162850830092274809642552, −4.15329548818133795366659552957, −1.91990139970199253497165291574, −1.10942129075621914336572047531, 1.21408212921249004983641022610, 2.45083073760314497510382976767, 2.93307882021924875828856441562, 5.00524884488957101275003455694, 5.94376498117900582618333508898, 6.83363748624356175548971420612, 7.956157244759833759075332600739, 8.869560331923992909323353653200, 9.462215316407101989732657910377, 10.04998901657220234170564565453

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