Properties

Label 2-1045-1045.949-c0-0-4
Degree $2$
Conductor $1045$
Sign $0.958 - 0.286i$
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  + (0.734 − 0.533i)2-s + (0.0966 + 0.297i)3-s + (−0.0542 + 0.166i)4-s + (0.809 + 0.587i)5-s + (0.229 + 0.166i)6-s + (0.329 + 1.01i)8-s + (0.729 − 0.530i)9-s + 0.907·10-s + (−0.587 − 0.809i)11-s − 0.0549·12-s + (−1.44 + 1.04i)13-s + (−0.0966 + 0.297i)15-s + (0.642 + 0.466i)16-s + (0.253 − 0.779i)18-s + (−0.309 − 0.951i)19-s + (−0.142 + 0.103i)20-s + ⋯
L(s)  = 1  + (0.734 − 0.533i)2-s + (0.0966 + 0.297i)3-s + (−0.0542 + 0.166i)4-s + (0.809 + 0.587i)5-s + (0.229 + 0.166i)6-s + (0.329 + 1.01i)8-s + (0.729 − 0.530i)9-s + 0.907·10-s + (−0.587 − 0.809i)11-s − 0.0549·12-s + (−1.44 + 1.04i)13-s + (−0.0966 + 0.297i)15-s + (0.642 + 0.466i)16-s + (0.253 − 0.779i)18-s + (−0.309 − 0.951i)19-s + (−0.142 + 0.103i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.958 - 0.286i$
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: \((2,\ 1045,\ (\ :0),\ 0.958 - 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.643101057\)
\(L(\frac12)\) \(\approx\) \(1.643101057\)
\(L(1)\) 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 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.734 + 0.533i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.44 - 1.04i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-1.14 + 0.831i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.97T + T^{2} \)
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 \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \)
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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

−10.24915409047901973552985351147, −9.449551208160673964482334476597, −8.749832614518633642499201988673, −7.46149413586005343390604403318, −6.81211995679277398912824190042, −5.64319952195481479055228626365, −4.78499724368508138693845332942, −3.92751319836873019523918792771, −2.84700666374357680077063875985, −2.11020156993529458127967562879, 1.48741450293689389947942293237, 2.66271274303901623638327362446, 4.38638379392891753280936826622, 4.95441038781820543069043894404, 5.63520522413502873177959110931, 6.60372872744355819898943629664, 7.48898379766910957493219183422, 8.124307458802593336279684151455, 9.522899260302812700977636822286, 10.10674000536615267471312605161

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