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Properties

Label 2-819-91.81-c1-0-38
Degree $2$
Conductor $819$
Sign $-0.982 + 0.188i$
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  + (1.02 − 1.77i)2-s + (−1.11 − 1.92i)4-s + (0.274 + 0.475i)5-s + (−0.839 − 2.50i)7-s − 0.456·8-s + 1.12·10-s − 4.69·11-s + (−0.663 − 3.54i)13-s + (−5.32 − 1.08i)14-s + (1.75 − 3.03i)16-s + (−0.301 − 0.522i)17-s + 0.561·19-s + (0.610 − 1.05i)20-s + (−4.82 + 8.36i)22-s + (0.188 − 0.326i)23-s + ⋯
L(s)  = 1  + (0.726 − 1.25i)2-s + (−0.555 − 0.962i)4-s + (0.122 + 0.212i)5-s + (−0.317 − 0.948i)7-s − 0.161·8-s + 0.356·10-s − 1.41·11-s + (−0.184 − 0.982i)13-s + (−1.42 − 0.289i)14-s + (0.438 − 0.759i)16-s + (−0.0731 − 0.126i)17-s + 0.128·19-s + (0.136 − 0.236i)20-s + (−1.02 + 1.78i)22-s + (0.0392 − 0.0680i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.982 + 0.188i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175878 - 1.85139i\)
\(L(\frac12)\) \(\approx\) \(0.175878 - 1.85139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.839 + 2.50i)T \)
13 \( 1 + (0.663 + 3.54i)T \)
good2 \( 1 + (-1.02 + 1.77i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.274 - 0.475i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.69T + 11T^{2} \)
17 \( 1 + (0.301 + 0.522i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.561T + 19T^{2} \)
23 \( 1 + (-0.188 + 0.326i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.09 + 3.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.577 - 0.999i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.40 + 7.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.96 - 6.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.747 - 1.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.09 - 1.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.52 - 7.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.26 - 7.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.42T + 61T^{2} \)
67 \( 1 + 9.59T + 67T^{2} \)
71 \( 1 + (2.88 - 5.00i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.24 + 12.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.31 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 + (-4.59 + 7.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.15 + 5.45i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.32401877367797961340958269026, −9.459719156104905433994010086654, −7.929813990503748529609225933974, −7.44179321474122483415030679600, −6.05746992527417074020762365687, −5.05322669095421240700934417876, −4.20136156389992630931976471209, −3.11175670966922389897863016228, −2.42205530628917753960023504827, −0.70913159138250785947221616637, 2.12388897658819517734965265929, 3.48153107763597668458784171799, 4.85464478273815895063779873374, 5.31304163868225951967914229676, 6.20702845329514875429481343777, 7.03335510678941035950313359853, 7.890376712054835840527855349927, 8.715082889146805358231521900397, 9.557002643666358464447266826009, 10.61216943684602838461550788219

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