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Properties

Label 2-819-117.2-c1-0-26
Degree $2$
Conductor $819$
Sign $-0.750 + 0.661i$
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  + (1.77 + 1.77i)2-s + (−0.654 + 1.60i)3-s + 4.29i·4-s + (−0.632 + 2.35i)5-s + (−4.00 + 1.68i)6-s + (0.258 − 0.965i)7-s + (−4.07 + 4.07i)8-s + (−2.14 − 2.10i)9-s + (−5.30 + 3.06i)10-s + (2.66 − 2.66i)11-s + (−6.88 − 2.81i)12-s + (−2.20 + 2.85i)13-s + (2.17 − 1.25i)14-s + (−3.36 − 2.55i)15-s − 5.85·16-s + (0.330 − 0.572i)17-s + ⋯
L(s)  = 1  + (1.25 + 1.25i)2-s + (−0.378 + 0.925i)3-s + 2.14i·4-s + (−0.282 + 1.05i)5-s + (−1.63 + 0.686i)6-s + (0.0978 − 0.365i)7-s + (−1.43 + 1.43i)8-s + (−0.714 − 0.700i)9-s + (−1.67 + 0.968i)10-s + (0.803 − 0.803i)11-s + (−1.98 − 0.811i)12-s + (−0.611 + 0.791i)13-s + (0.580 − 0.335i)14-s + (−0.869 − 0.660i)15-s − 1.46·16-s + (0.0801 − 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.750 + 0.661i$
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: \((2,\ 819,\ (\ :1/2),\ -0.750 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.813469 - 2.15406i\)
\(L(\frac12)\) \(\approx\) \(0.813469 - 2.15406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.654 - 1.60i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (2.20 - 2.85i)T \)
good2 \( 1 + (-1.77 - 1.77i)T + 2iT^{2} \)
5 \( 1 + (0.632 - 2.35i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.66 + 2.66i)T - 11iT^{2} \)
17 \( 1 + (-0.330 + 0.572i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.747 - 2.79i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.95 - 6.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.15iT - 29T^{2} \)
31 \( 1 + (-1.03 - 0.277i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.10 + 4.10i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-6.40 + 1.71i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.48 + 3.16i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.530 - 1.98i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 8.27iT - 53T^{2} \)
59 \( 1 + (4.97 - 4.97i)T - 59iT^{2} \)
61 \( 1 + (-4.94 - 8.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.06 + 7.70i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (14.2 - 3.82i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.40 + 2.40i)T + 73iT^{2} \)
79 \( 1 + (1.56 - 2.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.134 + 0.0360i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (3.19 + 0.856i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.3 - 3.31i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.97396144897067274890670112319, −9.905372401742223344313162794804, −8.973988089944022555303850572058, −7.72751685133668993840533424269, −7.14388848438335200342082784030, −6.07515121694391893909934563490, −5.74788836820977378479761739010, −4.31378141195410491010195507247, −3.93334079653557283142761154393, −2.96619283232308485938182666124, 0.818755508154503300377678545108, 1.92119450478476581376879290881, 2.95179757358722091222457863125, 4.44535522351994020774991157306, 4.93193379798744907285525795516, 5.84726725847084170903338024544, 6.82621480255799001856199006347, 8.040088714972510171655023558541, 8.994033925372171766514268096914, 10.03640311873789382546860062439

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