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Properties

Label 2-819-91.45-c1-0-3
Degree $2$
Conductor $819$
Sign $-0.907 + 0.420i$
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  + (−0.490 + 0.490i)2-s + 1.51i·4-s + (0.00962 − 0.0359i)5-s + (−0.176 + 2.63i)7-s + (−1.72 − 1.72i)8-s + (0.0129 + 0.0223i)10-s + (−0.292 + 1.09i)11-s + (−3.58 − 0.400i)13-s + (−1.20 − 1.38i)14-s − 1.33·16-s − 6.40·17-s + (3.56 − 0.954i)19-s + (0.0545 + 0.0146i)20-s + (−0.391 − 0.678i)22-s − 2.79i·23-s + ⋯
L(s)  = 1  + (−0.347 + 0.347i)2-s + 0.758i·4-s + (0.00430 − 0.0160i)5-s + (−0.0668 + 0.997i)7-s + (−0.610 − 0.610i)8-s + (0.00408 + 0.00707i)10-s + (−0.0880 + 0.328i)11-s + (−0.993 − 0.111i)13-s + (−0.323 − 0.369i)14-s − 0.334·16-s − 1.55·17-s + (0.816 − 0.218i)19-s + (0.0121 + 0.00326i)20-s + (−0.0835 − 0.144i)22-s − 0.582i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.907 + 0.420i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.907 + 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0976778 - 0.443444i\)
\(L(\frac12)\) \(\approx\) \(0.0976778 - 0.443444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.176 - 2.63i)T \)
13 \( 1 + (3.58 + 0.400i)T \)
good2 \( 1 + (0.490 - 0.490i)T - 2iT^{2} \)
5 \( 1 + (-0.00962 + 0.0359i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.292 - 1.09i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 6.40T + 17T^{2} \)
19 \( 1 + (-3.56 + 0.954i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 2.79iT - 23T^{2} \)
29 \( 1 + (1.84 - 3.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.63 - 0.706i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.94 + 3.94i)T + 37iT^{2} \)
41 \( 1 + (-0.188 + 0.0505i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.84 + 1.06i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.43 + 1.45i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.295 + 0.512i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.97 - 7.97i)T - 59iT^{2} \)
61 \( 1 + (1.18 + 0.686i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.28 + 1.95i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.88 - 2.64i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.707 - 2.64i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.63 + 2.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.92 - 4.92i)T + 83iT^{2} \)
89 \( 1 + (11.9 - 11.9i)T - 89iT^{2} \)
97 \( 1 + (0.663 - 2.47i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.73338561315974954037880660511, −9.444107516988485537969578239123, −9.062120719480424345522801057011, −8.246725972488423907674591254272, −7.22870039163634374974249312033, −6.70195210134206502475087422018, −5.43040559609514381498709161570, −4.51788581392623805148157113238, −3.15657160093014402768369969143, −2.25032699814187519128197990011, 0.23742461585932232569034735963, 1.69629355917625439064409523230, 2.97268864858376936631964773177, 4.37947302950748935684274188251, 5.19793941472744238171153137070, 6.36148528525604937222520167831, 7.11484843509072385696024810411, 8.121904193809199549142034887629, 9.211923067115254755020177348805, 9.760610501784782692319643563834

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