Properties

Label 2-819-7.4-c1-0-25
Degree $2$
Conductor $819$
Sign $0.968 + 0.250i$
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  + (0.5 + 0.866i)2-s + (0.500 − 0.866i)4-s + (0.5 − 2.59i)7-s + 3·8-s + (−1.5 + 2.59i)11-s − 13-s + (2.5 − 0.866i)14-s + (0.500 + 0.866i)16-s + (3.5 − 6.06i)17-s + (3.5 + 6.06i)19-s − 3·22-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s + (−0.5 − 0.866i)26-s + (−2 − 1.73i)28-s + 5·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.250 − 0.433i)4-s + (0.188 − 0.981i)7-s + 1.06·8-s + (−0.452 + 0.783i)11-s − 0.277·13-s + (0.668 − 0.231i)14-s + (0.125 + 0.216i)16-s + (0.848 − 1.47i)17-s + (0.802 + 1.39i)19-s − 0.639·22-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + (−0.0980 − 0.169i)26-s + (−0.377 − 0.327i)28-s + 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12441 - 0.270721i\)
\(L(\frac12)\) \(\approx\) \(2.12441 - 0.270721i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.20705804844234247627631597015, −9.649093909263032705558068898876, −8.115862681939855909508179202790, −7.44277587955183467098704020207, −6.86628225970989970342813747361, −5.76145459283401314157428205993, −4.91294070431899775976157886807, −4.12992419497508574910374075878, −2.56480554037620395877932443742, −1.05941418290059272872830620394, 1.61074468775497046483449544064, 2.85778785863875008767717600319, 3.51361928136713848133171515249, 4.90515130100010930980228173606, 5.65876930971662911685572106022, 6.82243445376342445908359488603, 7.893982136723309033224020138270, 8.453963849240288350915207702426, 9.503742254310624357392569201554, 10.50062690503833975411734451573

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