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Properties

Label 2-819-91.51-c1-0-26
Degree $2$
Conductor $819$
Sign $-0.400 + 0.916i$
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  + (−0.287 − 0.166i)2-s + (−0.944 − 1.63i)4-s + (−1.25 − 0.722i)5-s + (2.26 − 1.36i)7-s + 1.29i·8-s + (0.240 + 0.416i)10-s + (5.15 − 2.97i)11-s + (1.88 + 3.07i)13-s + (−0.879 + 0.0178i)14-s + (−1.67 + 2.90i)16-s + (−2.16 − 3.74i)17-s + (1.69 + 0.978i)19-s + 2.73i·20-s − 1.97·22-s + (0.270 − 0.467i)23-s + ⋯
L(s)  = 1  + (−0.203 − 0.117i)2-s + (−0.472 − 0.818i)4-s + (−0.559 − 0.323i)5-s + (0.855 − 0.517i)7-s + 0.457i·8-s + (0.0759 + 0.131i)10-s + (1.55 − 0.897i)11-s + (0.524 + 0.851i)13-s + (−0.234 + 0.00477i)14-s + (−0.418 + 0.725i)16-s + (−0.524 − 0.909i)17-s + (0.388 + 0.224i)19-s + 0.610i·20-s − 0.421·22-s + (0.0563 − 0.0975i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.400 + 0.916i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.400 + 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650870 - 0.994606i\)
\(L(\frac12)\) \(\approx\) \(0.650870 - 0.994606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.26 + 1.36i)T \)
13 \( 1 + (-1.88 - 3.07i)T \)
good2 \( 1 + (0.287 + 0.166i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.25 + 0.722i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.15 + 2.97i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.69 - 0.978i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.270 + 0.467i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.15T + 29T^{2} \)
31 \( 1 + (-5.28 + 3.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.95 + 4.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.55iT - 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 + (-5.42 - 3.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.38 + 2.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.737 + 0.425i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.854 + 0.493i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.76iT - 71T^{2} \)
73 \( 1 + (7.91 - 4.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.0655 + 0.113i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.66iT - 83T^{2} \)
89 \( 1 + (-8.41 - 4.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.58iT - 97T^{2} \)
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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

−9.889491955640076441580454614249, −8.915751198352495747963652577384, −8.663160732481641179249115285508, −7.43993262565476887677136155757, −6.46209428923613144877288549456, −5.48465506237575363255646787764, −4.38112803298947676286412483116, −3.84713590112824073837922205666, −1.78705532386095804571140797055, −0.70837654730791155110772668458, 1.60457779126785139087215755303, 3.27389273451922093159912607953, 4.06651960853159784948612610507, 4.98459675605520256371068225742, 6.32401275208980368267578134437, 7.26899403162922136990043111721, 7.998415602373728364247438146764, 8.743614157004217496679430303940, 9.392951036230421696897921123519, 10.52035263650930205626135625592

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