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Properties

Label 2-805-7.4-c1-0-9
Degree $2$
Conductor $805$
Sign $0.243 - 0.969i$
Analytic cond. $6.42795$
Root an. cond. $2.53534$
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  + (−1.39 − 2.41i)2-s + (−0.706 + 1.22i)3-s + (−2.88 + 5.00i)4-s + (0.5 + 0.866i)5-s + 3.94·6-s + (0.884 + 2.49i)7-s + 10.5·8-s + (0.500 + 0.866i)9-s + (1.39 − 2.41i)10-s + (−1.91 + 3.31i)11-s + (−4.08 − 7.07i)12-s + 3.24·13-s + (4.78 − 5.61i)14-s − 1.41·15-s + (−8.92 − 15.4i)16-s + (−1.39 + 2.41i)17-s + ⋯
L(s)  = 1  + (−0.986 − 1.70i)2-s + (−0.408 + 0.706i)3-s + (−1.44 + 2.50i)4-s + (0.223 + 0.387i)5-s + 1.61·6-s + (0.334 + 0.942i)7-s + 3.72·8-s + (0.166 + 0.288i)9-s + (0.441 − 0.763i)10-s + (−0.577 + 1.00i)11-s + (−1.17 − 2.04i)12-s + 0.901·13-s + (1.28 − 1.50i)14-s − 0.365·15-s + (−2.23 − 3.86i)16-s + (−0.337 + 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(6.42795\)
Root analytic conductor: \(2.53534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :1/2),\ 0.243 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.485731 + 0.378785i\)
\(L(\frac12)\) \(\approx\) \(0.485731 + 0.378785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.884 - 2.49i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.39 + 2.41i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.706 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.91 - 3.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
17 \( 1 + (1.39 - 2.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.05 - 1.82i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 8.56T + 29T^{2} \)
31 \( 1 + (-4.43 + 7.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.231 - 0.400i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + (-4.30 - 7.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.06 - 5.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.72 + 4.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.57 + 4.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.70 + 9.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 + (3.52 - 6.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.04 + 1.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.64T + 83T^{2} \)
89 \( 1 + (6.96 + 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.95T + 97T^{2} \)
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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

−10.57416224817251008650078364155, −9.681838419601752001810235519506, −9.263853216321602036261414217779, −8.152180406590936951402347389331, −7.55194456408691119104217803010, −5.80277766497333654287482138665, −4.62722740820717510287843180181, −3.85179824896886593404879568137, −2.52297853616862356363226477956, −1.75875807236450304081260217712, 0.50502696093082656554644123374, 1.35837017230361688998921446570, 4.07709236383398229557083054000, 5.30098381115557798530397007661, 5.90086070082327506879043175218, 6.86245519654394594631861582243, 7.34166129553553936758532967026, 8.255228223110091563178414993403, 8.902262343997571195707359203094, 9.784428190224829769394659422017

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