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Properties

Label 2-819-91.74-c1-0-36
Degree $2$
Conductor $819$
Sign $0.991 - 0.132i$
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  + 2.43·2-s + 3.92·4-s + (0.613 + 1.06i)5-s + (2.20 − 1.46i)7-s + 4.68·8-s + (1.49 + 2.58i)10-s + (1.74 + 3.02i)11-s + (−2.87 − 2.17i)13-s + (5.35 − 3.57i)14-s + 3.55·16-s − 4.52·17-s + (−0.677 + 1.17i)19-s + (2.40 + 4.17i)20-s + (4.25 + 7.36i)22-s + 0.673·23-s + ⋯
L(s)  = 1  + 1.72·2-s + 1.96·4-s + (0.274 + 0.475i)5-s + (0.831 − 0.554i)7-s + 1.65·8-s + (0.472 + 0.818i)10-s + (0.526 + 0.911i)11-s + (−0.797 − 0.603i)13-s + (1.43 − 0.954i)14-s + 0.888·16-s − 1.09·17-s + (−0.155 + 0.269i)19-s + (0.538 + 0.933i)20-s + (0.906 + 1.56i)22-s + 0.140·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.991 - 0.132i$
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: \((2,\ 819,\ (\ :1/2),\ 0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.47549 + 0.297296i\)
\(L(\frac12)\) \(\approx\) \(4.47549 + 0.297296i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 1.46i)T \)
13 \( 1 + (2.87 + 2.17i)T \)
good2 \( 1 - 2.43T + 2T^{2} \)
5 \( 1 + (-0.613 - 1.06i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.74 - 3.02i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 + (0.677 - 1.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.673T + 23T^{2} \)
29 \( 1 + (2.64 - 4.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.99 + 8.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.08T + 37T^{2} \)
41 \( 1 + (3.61 - 6.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.48 - 7.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.58 + 4.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.95 + 8.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.803T + 59T^{2} \)
61 \( 1 + (2.32 - 4.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.06 - 1.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.52 - 4.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.04 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.90 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (3.59 + 6.22i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53186240754998124013411046538, −9.721255421259712619404428913121, −8.292732687324248233383961140501, −7.18700195303634675964156779693, −6.71330017940389703399153355659, −5.64860327275893829648436999983, −4.65282183536960826801077236451, −4.19595992681010281472348263402, −2.85322456826963322955635879248, −1.90527659181437268732762395719, 1.77755593261791955218931379751, 2.80937062956901233652855834075, 4.08637294010338376219162271916, 4.85823763844532304911867294943, 5.51790556811271048649968036947, 6.44047155783803627986230157971, 7.26105470175440864692586060313, 8.630336632123181810574956725750, 9.161714425913537558691705472007, 10.69270894370580237922295991087

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