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Properties

Label 2-819-117.2-c1-0-53
Degree $2$
Conductor $819$
Sign $-0.547 - 0.837i$
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  + (−1.91 − 1.91i)2-s + (−1.50 − 0.863i)3-s + 5.31i·4-s + (0.990 − 3.69i)5-s + (1.22 + 4.52i)6-s + (0.258 − 0.965i)7-s + (6.34 − 6.34i)8-s + (1.50 + 2.59i)9-s + (−8.96 + 5.17i)10-s + (−0.392 + 0.392i)11-s + (4.59 − 7.98i)12-s + (3.10 + 1.83i)13-s + (−2.34 + 1.35i)14-s + (−4.68 + 4.69i)15-s − 13.6·16-s + (1.77 − 3.07i)17-s + ⋯
L(s)  = 1  + (−1.35 − 1.35i)2-s + (−0.866 − 0.498i)3-s + 2.65i·4-s + (0.443 − 1.65i)5-s + (0.498 + 1.84i)6-s + (0.0978 − 0.365i)7-s + (2.24 − 2.24i)8-s + (0.502 + 0.864i)9-s + (−2.83 + 1.63i)10-s + (−0.118 + 0.118i)11-s + (1.32 − 2.30i)12-s + (0.861 + 0.508i)13-s + (−0.626 + 0.361i)14-s + (−1.20 + 1.21i)15-s − 3.40·16-s + (0.430 − 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.547 - 0.837i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.547 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248201 + 0.458700i\)
\(L(\frac12)\) \(\approx\) \(0.248201 + 0.458700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.50 + 0.863i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-3.10 - 1.83i)T \)
good2 \( 1 + (1.91 + 1.91i)T + 2iT^{2} \)
5 \( 1 + (-0.990 + 3.69i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.392 - 0.392i)T - 11iT^{2} \)
17 \( 1 + (-1.77 + 3.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.43 + 5.36i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.405 + 0.702i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.85iT - 29T^{2} \)
31 \( 1 + (-3.81 - 1.02i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.22 + 4.57i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (6.56 - 1.76i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.371 - 0.214i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.80 - 10.4i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 5.57iT - 53T^{2} \)
59 \( 1 + (-0.279 + 0.279i)T - 59iT^{2} \)
61 \( 1 + (1.72 + 2.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.14 - 8.01i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (15.0 - 4.04i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.85 - 1.85i)T + 73iT^{2} \)
79 \( 1 + (-1.53 + 2.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.42 + 1.99i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-16.0 - 4.31i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-13.1 - 3.52i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.694034212449238636088912703333, −9.035820968792426164571959781562, −8.258228126337673687763417255790, −7.50840486864897543621137352769, −6.32970733856997384402289349893, −4.91946015899747546951856545688, −4.16718490010439099816040030617, −2.36941652973999852343803242303, −1.28221909804843899109355572099, −0.52865465417865744950791421709, 1.55874753783130290521415355576, 3.46088182338201334635093356197, 5.23239533133490907650012024681, 6.01664928280384427939413023433, 6.39039507182083663002279973325, 7.23912270562082751978320135926, 8.187245960603547118906568508396, 9.070398093656589734542100456056, 10.21648778741990126408963779689, 10.36066637142886173957200210630

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