Properties

Label 2-1045-1045.417-c0-0-0
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: $\chi_{1045} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6145859214\)
\(L(\frac12)\) \(\approx\) \(0.6145859214\)
\(L(1)\) 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

−9.693456368266116561304895484405, −9.190234876684105935306729063950, −8.333479885990558255116293997436, −7.85204055585456626447240482814, −6.19626198096216368649407430673, −5.68826381795151903456043795518, −4.05035672074264593152838961880, −3.42725120732277916088107447536, −1.79926170213146696960084929396, −0.868387995510878913450839233923, 1.75206183071507496831304329491, 3.24554637044310628553852979076, 4.40768063736441103055922973967, 5.77642914467062816249419478091, 6.75263552876485218216078419294, 6.88366514467458759264822792460, 8.013869389305407121251535938426, 8.707675254976155888493077512234, 9.397071014794155632236721369133, 10.33475460401745601792993923074

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