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Properties

Label 2-819-91.9-c1-0-32
Degree $2$
Conductor $819$
Sign $-0.0787 + 0.996i$
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  + (−0.415 − 0.719i)2-s + (0.654 − 1.13i)4-s + (1.30 − 2.26i)5-s + (0.801 + 2.52i)7-s − 2.75·8-s − 2.17·10-s + 1.84·11-s + (2.74 + 2.33i)13-s + (1.48 − 1.62i)14-s + (−0.165 − 0.287i)16-s + (3.41 − 5.91i)17-s + 5.06·19-s + (−1.71 − 2.96i)20-s + (−0.768 − 1.33i)22-s + (−0.636 − 1.10i)23-s + ⋯
L(s)  = 1  + (−0.293 − 0.509i)2-s + (0.327 − 0.566i)4-s + (0.585 − 1.01i)5-s + (0.302 + 0.952i)7-s − 0.972·8-s − 0.687·10-s + 0.557·11-s + (0.761 + 0.648i)13-s + (0.396 − 0.434i)14-s + (−0.0414 − 0.0717i)16-s + (0.828 − 1.43i)17-s + 1.16·19-s + (−0.382 − 0.663i)20-s + (−0.163 − 0.283i)22-s + (−0.132 − 0.229i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0787 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0787 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0787 + 0.996i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.0787 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17023 - 1.26633i\)
\(L(\frac12)\) \(\approx\) \(1.17023 - 1.26633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.801 - 2.52i)T \)
13 \( 1 + (-2.74 - 2.33i)T \)
good2 \( 1 + (0.415 + 0.719i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.30 + 2.26i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.84T + 11T^{2} \)
17 \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 5.06T + 19T^{2} \)
23 \( 1 + (0.636 + 1.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.724 - 1.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.09 + 5.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.93 + 6.82i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.41 - 7.64i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.109 - 0.189i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.624 - 1.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.33 + 2.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.01 - 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.72T + 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + (1.78 + 3.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.26 + 5.65i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.08 + 5.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.67T + 83T^{2} \)
89 \( 1 + (7.57 + 13.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.08 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.699020348456915161977197414962, −9.295260653318341359493993382033, −8.797562869817997595430615164705, −7.51974604678361587683150156148, −6.27972311328209857398169168305, −5.54844023460980647450732670098, −4.87591786164611999317121649178, −3.23629158520340910180662952534, −1.96505401548148924666114132088, −1.07768804512887670531943938517, 1.56481994745208551397946564599, 3.22651066981977656054744169782, 3.70965378981640912550412480860, 5.46272500309048999685298575709, 6.38138761751997616672234227993, 6.98863751128037235392100615411, 7.83330646235696137148357791555, 8.500694514127120130898903569826, 9.696483931724792587780124614847, 10.47627029902893491737412090277

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