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Properties

Label 2-819-91.74-c1-0-44
Degree $2$
Conductor $819$
Sign $-0.964 + 0.265i$
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  + 1.71·2-s + 0.941·4-s + (−1.22 − 2.12i)5-s + (−2.38 + 1.14i)7-s − 1.81·8-s + (−2.10 − 3.64i)10-s + (−0.519 − 0.899i)11-s + (−3.36 − 1.29i)13-s + (−4.09 + 1.95i)14-s − 4.99·16-s − 3.01·17-s + (−1.59 + 2.76i)19-s + (−1.15 − 2.00i)20-s + (−0.890 − 1.54i)22-s + 3.47·23-s + ⋯
L(s)  = 1  + 1.21·2-s + 0.470·4-s + (−0.549 − 0.951i)5-s + (−0.902 + 0.431i)7-s − 0.641·8-s + (−0.666 − 1.15i)10-s + (−0.156 − 0.271i)11-s + (−0.932 − 0.360i)13-s + (−1.09 + 0.523i)14-s − 1.24·16-s − 0.732·17-s + (−0.366 + 0.634i)19-s + (−0.258 − 0.447i)20-s + (−0.189 − 0.328i)22-s + 0.723·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0836295 - 0.617801i\)
\(L(\frac12)\) \(\approx\) \(0.0836295 - 0.617801i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.38 - 1.14i)T \)
13 \( 1 + (3.36 + 1.29i)T \)
good2 \( 1 - 1.71T + 2T^{2} \)
5 \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.519 + 0.899i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.01T + 17T^{2} \)
19 \( 1 + (1.59 - 2.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (-4.01 + 6.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + (-2.54 + 4.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.21 + 5.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.88 + 8.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.90 - 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + (-1.30 + 2.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.61 - 9.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.63 - 2.82i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.50 + 13.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.211 + 0.366i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.34T + 83T^{2} \)
89 \( 1 + 5.30T + 89T^{2} \)
97 \( 1 + (-2.92 - 5.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.761700986869411361031594146988, −8.946574026265275435314284293445, −8.262421808888484011349903893149, −6.99582372913608150762489091940, −6.07618529352218508515673871801, −5.20165996648898392502833681785, −4.48746370113037555469431962433, −3.52389982811844557882125515696, −2.50131607667951562138488919988, −0.19129626980742535521250403823, 2.61230316777650773774839654536, 3.28257080337879066167129454924, 4.30858101017977868581157907718, 5.04543351357779394251838031149, 6.50702753176999314969104733945, 6.73953477145612118638027828241, 7.73984451165325673358805181143, 9.137020534323291377656471224223, 9.779232104253022687807812983761, 10.99582377158371048720102741761

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