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Properties

Label 2-819-117.88-c1-0-41
Degree $2$
Conductor $819$
Sign $0.325 - 0.945i$
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  + 1.68i·2-s + (1.44 − 0.948i)3-s − 0.851·4-s + (1.72 − 0.997i)5-s + (1.60 + 2.44i)6-s + (−0.866 + 0.5i)7-s + 1.93i·8-s + (1.20 − 2.74i)9-s + (1.68 + 2.91i)10-s + 2.78i·11-s + (−1.23 + 0.807i)12-s + (0.954 + 3.47i)13-s + (−0.844 − 1.46i)14-s + (1.55 − 3.08i)15-s − 4.97·16-s + (1.71 − 2.97i)17-s + ⋯
L(s)  = 1  + 1.19i·2-s + (0.836 − 0.547i)3-s − 0.425·4-s + (0.773 − 0.446i)5-s + (0.653 + 0.999i)6-s + (−0.327 + 0.188i)7-s + 0.685i·8-s + (0.400 − 0.916i)9-s + (0.532 + 0.923i)10-s + 0.840i·11-s + (−0.356 + 0.233i)12-s + (0.264 + 0.964i)13-s + (−0.225 − 0.390i)14-s + (0.402 − 0.796i)15-s − 1.24·16-s + (0.416 − 0.721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.325 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00825 + 1.43267i\)
\(L(\frac12)\) \(\approx\) \(2.00825 + 1.43267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.44 + 0.948i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.954 - 3.47i)T \)
good2 \( 1 - 1.68iT - 2T^{2} \)
5 \( 1 + (-1.72 + 0.997i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.78iT - 11T^{2} \)
17 \( 1 + (-1.71 + 2.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.55 - 1.47i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.434 + 0.751i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.47T + 29T^{2} \)
31 \( 1 + (-5.38 + 3.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.93 + 3.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.95 + 1.12i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.31 + 2.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.45 - 2.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.32T + 53T^{2} \)
59 \( 1 + 12.5iT - 59T^{2} \)
61 \( 1 + (-3.33 - 5.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.81 + 1.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.10 + 4.68i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + (-1.52 + 2.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.1 + 5.83i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (15.1 - 8.77i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.82 + 3.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.776642141487635811116245984418, −9.416378974832894079778080959063, −8.583341074646156267297384508562, −7.66330978438160979747164330094, −7.02639774055643046476341675250, −6.23573625799950336865868826570, −5.39695246853418941654762077252, −4.24068826402883735810700521419, −2.67638235913217761508813388221, −1.65035511718297571526056142297, 1.32039037003552382391252309192, 2.72560816506616126759856336862, 3.15697414070870668072362457883, 4.15521892548875554535548591711, 5.55228263364995549545234627043, 6.52964371441862896403741076776, 7.73452935001853406013128852562, 8.656712184774621257478939654549, 9.615327373529202101766295262232, 10.16058443790969106911359856273

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