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Properties

Label 2-819-91.59-c1-0-30
Degree $2$
Conductor $819$
Sign $-0.995 + 0.0994i$
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  + (−1.51 − 1.51i)2-s + 2.58i·4-s + (1.34 + 0.360i)5-s + (−0.246 − 2.63i)7-s + (0.893 − 0.893i)8-s + (−1.49 − 2.58i)10-s + (−0.336 − 0.0902i)11-s + (−1.32 − 3.35i)13-s + (−3.61 + 4.36i)14-s + 2.47·16-s + 0.982·17-s + (1.19 + 4.45i)19-s + (−0.933 + 3.48i)20-s + (0.373 + 0.646i)22-s − 3.30i·23-s + ⋯
L(s)  = 1  + (−1.07 − 1.07i)2-s + 1.29i·4-s + (0.601 + 0.161i)5-s + (−0.0931 − 0.995i)7-s + (0.315 − 0.315i)8-s + (−0.471 − 0.816i)10-s + (−0.101 − 0.0272i)11-s + (−0.368 − 0.929i)13-s + (−0.966 + 1.16i)14-s + 0.618·16-s + 0.238·17-s + (0.273 + 1.02i)19-s + (−0.208 + 0.778i)20-s + (0.0796 + 0.137i)22-s − 0.689i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.995 + 0.0994i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.995 + 0.0994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0337356 - 0.677016i\)
\(L(\frac12)\) \(\approx\) \(0.0337356 - 0.677016i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.246 + 2.63i)T \)
13 \( 1 + (1.32 + 3.35i)T \)
good2 \( 1 + (1.51 + 1.51i)T + 2iT^{2} \)
5 \( 1 + (-1.34 - 0.360i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.336 + 0.0902i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.982T + 17T^{2} \)
19 \( 1 + (-1.19 - 4.45i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 3.30iT - 23T^{2} \)
29 \( 1 + (-0.941 + 1.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.755 + 2.81i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \)
41 \( 1 + (0.580 + 2.16i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.47 - 3.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.83 + 10.5i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.77 - 6.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.7 + 10.7i)T + 59iT^{2} \)
61 \( 1 + (5.59 + 3.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.61 + 9.74i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.43 - 9.07i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-10.3 + 2.76i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.890 - 1.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.33 - 8.33i)T - 83iT^{2} \)
89 \( 1 + (-9.61 - 9.61i)T + 89iT^{2} \)
97 \( 1 + (12.6 + 3.39i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.907125745295508921439057202740, −9.423875707183058162265088755938, −8.098493654604514250389275249191, −7.75316674919871488703111856730, −6.42773922212949102486684828653, −5.41032040507764551541323013527, −3.95327633396132480052187245939, −2.92666796781546472424875232178, −1.80824220022601158150183485227, −0.48303034165563827500317116214, 1.57051610930892094969660190732, 2.98923707131198715958113562546, 4.79439668133955478718120099496, 5.69978817806826929319986089260, 6.42936128320748466817679523153, 7.26742347092195379057597640611, 8.145027929204587494684380320301, 9.132912797858279650723178489275, 9.341492737556656478408343086629, 10.16549180575806117017819468855

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