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Properties

Label 2-819-13.9-c1-0-35
Degree $2$
Conductor $819$
Sign $-0.0128 - 0.999i$
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 − 2.26i)2-s + (−2.42 − 4.20i)4-s − 2.61·5-s + (−0.5 − 0.866i)7-s − 7.47·8-s + (−3.42 + 5.93i)10-s + (0.927 − 1.60i)11-s + (−2.5 + 2.59i)13-s − 2.61·14-s + (−4.92 + 8.53i)16-s + (−0.736 − 1.27i)17-s + (0.927 + 1.60i)19-s + (6.35 + 11.0i)20-s + (−2.42 − 4.20i)22-s + (−2.23 + 3.87i)23-s + ⋯
L(s)  = 1  + (0.925 − 1.60i)2-s + (−1.21 − 2.10i)4-s − 1.17·5-s + (−0.188 − 0.327i)7-s − 2.64·8-s + (−1.08 + 1.87i)10-s + (0.279 − 0.484i)11-s + (−0.693 + 0.720i)13-s − 0.699·14-s + (−1.23 + 2.13i)16-s + (−0.178 − 0.309i)17-s + (0.212 + 0.368i)19-s + (1.42 + 2.46i)20-s + (−0.517 − 0.896i)22-s + (−0.466 + 0.807i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0128 - 0.999i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548828 + 0.555912i\)
\(L(\frac12)\) \(\approx\) \(0.548828 + 0.555912i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (2.5 - 2.59i)T \)
good2 \( 1 + (-1.30 + 2.26i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
11 \( 1 + (-0.927 + 1.60i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.736 + 1.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.927 - 1.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 - 3.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.54 + 6.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.381 + 0.661i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.28 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 + 3.76T + 53T^{2} \)
59 \( 1 + (1.11 + 1.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.35 + 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.09 + 12.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + (-2.45 + 4.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.42 + 16.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.890542231609911978436830836607, −9.140621626766697723416951088189, −7.992384227023759368066073584524, −6.90127326116260062128694394373, −5.67029187898615476869807206661, −4.61189025050259106713142911174, −3.89903283346645414979513275564, −3.20367588080204052474263187610, −1.86470705892950145719818766659, −0.27888452567664882608029777316, 2.95481393297346341216492280605, 3.96387311810405721481461993773, 4.73633806205797881762617082683, 5.57716746810252454261449397082, 6.67660867557620395054065618800, 7.25690390925433934590195057313, 8.073654403477034133434698744782, 8.646827848527413045738062189053, 9.758861278829956431781558097989, 11.07643165884477248768790389092

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