Properties

Label 2-819-91.54-c1-0-28
Degree $2$
Conductor $819$
Sign $0.738 + 0.674i$
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  + (1.30 − 1.30i)2-s − 1.42i·4-s + (0.0744 − 0.0199i)5-s + (2.55 + 0.700i)7-s + (0.758 + 0.758i)8-s + (0.0713 − 0.123i)10-s + (−0.672 + 0.180i)11-s + (2.39 + 2.69i)13-s + (4.25 − 2.42i)14-s + 4.82·16-s + 1.23·17-s + (−1.45 + 5.44i)19-s + (−0.0283 − 0.105i)20-s + (−0.643 + 1.11i)22-s − 7.37i·23-s + ⋯
L(s)  = 1  + (0.924 − 0.924i)2-s − 0.710i·4-s + (0.0333 − 0.00892i)5-s + (0.964 + 0.264i)7-s + (0.268 + 0.268i)8-s + (0.0225 − 0.0390i)10-s + (−0.202 + 0.0543i)11-s + (0.665 + 0.746i)13-s + (1.13 − 0.647i)14-s + 1.20·16-s + 0.300·17-s + (−0.334 + 1.24i)19-s + (−0.00633 − 0.0236i)20-s + (−0.137 + 0.237i)22-s − 1.53i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.738 + 0.674i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.738 + 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70834 - 1.05032i\)
\(L(\frac12)\) \(\approx\) \(2.70834 - 1.05032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.55 - 0.700i)T \)
13 \( 1 + (-2.39 - 2.69i)T \)
good2 \( 1 + (-1.30 + 1.30i)T - 2iT^{2} \)
5 \( 1 + (-0.0744 + 0.0199i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.672 - 0.180i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 1.23T + 17T^{2} \)
19 \( 1 + (1.45 - 5.44i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 7.37iT - 23T^{2} \)
29 \( 1 + (1.84 + 3.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.34 + 8.76i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.25 - 3.25i)T + 37iT^{2} \)
41 \( 1 + (-0.231 + 0.865i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.14 + 1.24i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.108 + 0.404i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.75 + 9.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.90 - 9.90i)T - 59iT^{2} \)
61 \( 1 + (-5.63 + 3.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.540 + 2.01i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.119 + 0.445i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.91 + 1.31i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.26 + 5.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.54 - 9.54i)T + 83iT^{2} \)
89 \( 1 + (3.14 - 3.14i)T - 89iT^{2} \)
97 \( 1 + (15.6 - 4.19i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.46412174500677123895547579571, −9.518069654529818187909586988245, −8.249808133596057809616448182781, −7.86370712960251686651895378079, −6.31996730022155573205953247665, −5.48231920010576889635557561973, −4.43734274460845579521989680296, −3.84871119787333536641430557192, −2.46107274973958927061237533394, −1.60023704647792548160198399588, 1.36892362497375209216035545085, 3.18018579155722137032247816395, 4.26608851559951802566328712439, 5.13135069190201083505417299906, 5.76968256448675886690865655140, 6.80178400514789462386434343086, 7.63734349822345568252397923057, 8.255277114970648762837855880778, 9.403451012355257842162476988914, 10.53074136328338853603260714370

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