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Properties

Label 2-819-7.4-c1-0-26
Degree $2$
Conductor $819$
Sign $0.839 + 0.543i$
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  + (0.632 + 1.09i)2-s + (0.199 − 0.344i)4-s + (−1.45 − 2.51i)5-s + (−1.29 + 2.30i)7-s + 3.03·8-s + (1.83 − 3.18i)10-s + (1.01 − 1.76i)11-s + 13-s + (−3.34 + 0.0400i)14-s + (1.52 + 2.63i)16-s + (1.99 − 3.46i)17-s + (−3.48 − 6.02i)19-s − 1.15·20-s + 2.57·22-s + (−0.313 − 0.543i)23-s + ⋯
L(s)  = 1  + (0.447 + 0.775i)2-s + (0.0995 − 0.172i)4-s + (−0.649 − 1.12i)5-s + (−0.489 + 0.871i)7-s + 1.07·8-s + (0.580 − 1.00i)10-s + (0.307 − 0.531i)11-s + 0.277·13-s + (−0.894 + 0.0107i)14-s + (0.380 + 0.659i)16-s + (0.484 − 0.839i)17-s + (−0.798 − 1.38i)19-s − 0.258·20-s + 0.549·22-s + (−0.0653 − 0.113i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.839 + 0.543i$
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: \((2,\ 819,\ (\ :1/2),\ 0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71501 - 0.507122i\)
\(L(\frac12)\) \(\approx\) \(1.71501 - 0.507122i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 - 2.30i)T \)
13 \( 1 - T \)
good2 \( 1 + (-0.632 - 1.09i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.45 + 2.51i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.01 + 1.76i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.99 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.48 + 6.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.313 + 0.543i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.09T + 29T^{2} \)
31 \( 1 + (-5.21 + 9.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.54 - 2.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.521T + 41T^{2} \)
43 \( 1 - 0.329T + 43T^{2} \)
47 \( 1 + (-5.27 - 9.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.55 + 6.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.01 + 1.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.20 + 2.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.34 - 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.60T + 71T^{2} \)
73 \( 1 + (1.48 - 2.57i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.38 - 7.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (1.34 + 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.32T + 97T^{2} \)
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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

−9.987282257899881552837389153207, −9.083433511620642476806258245666, −8.439157678244033447279288683645, −7.53588250021986858731365614554, −6.49187835912743851660382528897, −5.77911692211657670493915036912, −4.89342102473609770683152941253, −4.13651314501567067665908107288, −2.59168010083097474474629565582, −0.809282838100864788171264795646, 1.65232264000804205088481372714, 3.06830933350772790456096427651, 3.72942434799128095451469414932, 4.37246318825581131036046079279, 6.07168919984476019356246772985, 7.00424445258801971510824064188, 7.55269728010223677261657330389, 8.477991376295156317905947536635, 10.03220769237824258033554064560, 10.49826042406031651651390445337

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