Properties

Label 2-1045-1045.949-c0-0-5
Degree $2$
Conductor $1045$
Sign $0.286 + 0.958i$
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  + (−1.53 + 1.11i)2-s + (−0.363 − 1.11i)3-s + (0.809 − 2.48i)4-s + (−0.809 − 0.587i)5-s + (1.80 + 1.31i)6-s + (0.951 + 2.92i)8-s + (−0.309 + 0.224i)9-s + 1.90·10-s + (0.809 − 0.587i)11-s − 3.07·12-s + (1.53 − 1.11i)13-s + (−0.363 + 1.11i)15-s + (−2.61 − 1.90i)16-s + (0.224 − 0.690i)18-s + (0.309 + 0.951i)19-s + (−2.11 + 1.53i)20-s + ⋯
L(s)  = 1  + (−1.53 + 1.11i)2-s + (−0.363 − 1.11i)3-s + (0.809 − 2.48i)4-s + (−0.809 − 0.587i)5-s + (1.80 + 1.31i)6-s + (0.951 + 2.92i)8-s + (−0.309 + 0.224i)9-s + 1.90·10-s + (0.809 − 0.587i)11-s − 3.07·12-s + (1.53 − 1.11i)13-s + (−0.363 + 1.11i)15-s + (−2.61 − 1.90i)16-s + (0.224 − 0.690i)18-s + (0.309 + 0.951i)19-s + (−2.11 + 1.53i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4024669212\)
\(L(\frac12)\) \(\approx\) \(0.4024669212\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - 1.17T + T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.600982630043050337213174288406, −8.783725013556343934957782211171, −8.072675605173069013063847476450, −7.74542040187738281264754439318, −6.73220540186359521013115819523, −6.08891112844761003354731306522, −5.42428041034082298199854303362, −3.71153123224264895993674475358, −1.51129745340735722523562639872, −0.77320405199323260281190916169, 1.49648414052574090114986320280, 3.00527518065222213294863655595, 3.90959356959646672118857638629, 4.46084979734364782089507129783, 6.46454936074099107833956501188, 7.16753131619397321580552380384, 8.183781501472548031765011332587, 9.011763855758932671481951408294, 9.519724862622821140198105343276, 10.32462275630242761523685896718

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