Properties

Label 2-1045-1045.208-c0-0-3
Degree $2$
Conductor $1045$
Sign $0.850 + 0.525i$
Analytic cond. $0.521522$
Root an. cond. $0.722165$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s i·4-s + i·5-s + i·9-s + (1 + i)10-s + 11-s + (−1 − i)13-s + 16-s + (1 + i)18-s + i·19-s + 20-s + (1 − i)22-s + (−1 − i)23-s − 25-s − 2·26-s + ⋯
L(s)  = 1  + (1 − i)2-s i·4-s + i·5-s + i·9-s + (1 + i)10-s + 11-s + (−1 − i)13-s + 16-s + (1 + i)18-s + i·19-s + 20-s + (1 − i)22-s + (−1 − i)23-s − 25-s − 2·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.736474603\)
\(L(\frac12)\) \(\approx\) \(1.736474603\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
11 \( 1 - T \)
19 \( 1 - iT \)
good2 \( 1 + (-1 + i)T - iT^{2} \)
3 \( 1 - iT^{2} \)
7 \( 1 + iT^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1 - i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.18370987162743641083590069238, −9.854345473161023048654013660687, −8.119966507835763453819939323957, −7.68440188662252246586365449730, −6.40259518971876131588421356605, −5.60710096546892564815887618524, −4.53478897405041601349273461536, −3.75173983637425831397079934493, −2.69728799695975153235870854607, −1.98953361268774909305785756183, 1.50623806758096587503904121300, 3.46975110841508378527812456417, 4.30616758053099226860088634974, 4.98632450209736878733032099082, 5.89629164892603253984312620578, 6.78224605934323013604861424701, 7.27972448965396821559649864185, 8.512757720099854588943270119186, 9.292075785195061160221044845461, 9.808446244840076116337404848435

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