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Properties

Label 2-819-117.50-c1-0-52
Degree $2$
Conductor $819$
Sign $-0.655 - 0.754i$
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  + (−0.721 − 2.69i)2-s + (−1.27 + 1.17i)3-s + (−4.99 + 2.88i)4-s + (−0.822 − 3.06i)5-s + (4.08 + 2.57i)6-s + (0.707 − 0.707i)7-s + (7.41 + 7.41i)8-s + (0.228 − 2.99i)9-s + (−7.66 + 4.42i)10-s + (5.16 − 1.38i)11-s + (2.94 − 9.53i)12-s + (−1.12 − 3.42i)13-s + (−2.41 − 1.39i)14-s + (4.65 + 2.93i)15-s + (8.84 − 15.3i)16-s + (1.70 − 2.95i)17-s + ⋯
L(s)  = 1  + (−0.509 − 1.90i)2-s + (−0.733 + 0.679i)3-s + (−2.49 + 1.44i)4-s + (−0.367 − 1.37i)5-s + (1.66 + 1.04i)6-s + (0.267 − 0.267i)7-s + (2.62 + 2.62i)8-s + (0.0760 − 0.997i)9-s + (−2.42 + 1.40i)10-s + (1.55 − 0.417i)11-s + (0.851 − 2.75i)12-s + (−0.311 − 0.950i)13-s + (−0.644 − 0.372i)14-s + (1.20 + 0.756i)15-s + (2.21 − 3.83i)16-s + (0.413 − 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.655 - 0.754i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.655 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273179 + 0.599123i\)
\(L(\frac12)\) \(\approx\) \(0.273179 + 0.599123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.27 - 1.17i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (1.12 + 3.42i)T \)
good2 \( 1 + (0.721 + 2.69i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.822 + 3.06i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-5.16 + 1.38i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.70 + 2.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.235 - 0.0630i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 3.74T + 23T^{2} \)
29 \( 1 + (5.88 + 3.39i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.734 + 0.196i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.79 - 1.01i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.18 + 2.18i)T - 41iT^{2} \)
43 \( 1 + 3.05iT - 43T^{2} \)
47 \( 1 + (1.36 - 5.08i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 1.93iT - 53T^{2} \)
59 \( 1 + (1.35 - 5.04i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + (6.84 + 6.84i)T + 67iT^{2} \)
71 \( 1 + (-0.383 - 1.43i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.0490 - 0.0490i)T - 73iT^{2} \)
79 \( 1 + (-6.04 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.80 - 0.750i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-3.01 + 11.2i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.85 + 6.85i)T + 97iT^{2} \)
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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

−9.637371720503661594985632185987, −9.279179517190083132503302328456, −8.560561669080819064987565941357, −7.56732295094567200782521235892, −5.61736978546216953729940596284, −4.69736013502626674429038247028, −4.12462760397752438108328388972, −3.19670313026394365987479098957, −1.28291632304492642918752413560, −0.53955129401898711628254456184, 1.52422896015183185326395373924, 3.91479050801230848656490036862, 4.92983773749620403508436335801, 6.10743039585555916755629085799, 6.55863061338336999241555333204, 7.19232984679004947928297989591, 7.74657207837663548563590588865, 8.866076928007459418322509528685, 9.636789063236363506408420943587, 10.66072572869416831044130684611

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