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Properties

Label 2-1045-1045.664-c0-0-6
Degree $2$
Conductor $1045$
Sign $0.331 + 0.943i$
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  + (−0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (1.80 − 0.587i)7-s + (0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (−0.809 + 0.587i)16-s + (−0.690 − 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.309 + 0.951i)20-s − 1.17i·23-s + (0.309 + 0.951i)25-s + (−1.11 − 1.53i)28-s + (−1.80 − 0.587i)35-s + (0.309 − 0.951i)36-s + 1.17i·43-s + 0.999·44-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (1.80 − 0.587i)7-s + (0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (−0.809 + 0.587i)16-s + (−0.690 − 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.309 + 0.951i)20-s − 1.17i·23-s + (0.309 + 0.951i)25-s + (−1.11 − 1.53i)28-s + (−1.80 − 0.587i)35-s + (0.309 − 0.951i)36-s + 1.17i·43-s + 0.999·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.011225589\)
\(L(\frac12)\) \(\approx\) \(1.011225589\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.17iT - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
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 \( 1 + T^{2} \)
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 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−10.02960635231522908338192654219, −9.114832660421373135246338681309, −8.246901729422550879929990357377, −7.50719025279320017563468828050, −6.84197893935016528364593251708, −5.12876743556867126262837623481, −4.67497466134839374078109652980, −4.38912833790693122430251876095, −2.17532612982759790135681592626, −1.10269901584240731066200010138, 1.79974125629603895484407567238, 3.27676253265898877875472523749, 4.01018109915690361774245627949, 4.88277666701319250064485140216, 6.05309360147585951832195296621, 7.25824870409824046629302395546, 7.941692033838280110362612536553, 8.358708986598961914747556558341, 9.198608241933102518010538004359, 10.51716638731159475035146951768

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