Properties

Label 2-1045-1045.398-c0-0-0
Degree $2$
Conductor $1045$
Sign $0.998 - 0.0527i$
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  + (0.587 − 0.809i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 1.95i)7-s + (0.951 + 0.309i)9-s + (0.587 + 0.809i)11-s + (−0.309 − 0.951i)16-s + (−0.412 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.951 + 0.309i)20-s + (−1.26 − 1.26i)23-s + (0.309 + 0.951i)25-s + (1.76 + 0.896i)28-s + (0.896 − 1.76i)35-s + (0.809 − 0.587i)36-s + (1.26 − 1.26i)43-s + 44-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 1.95i)7-s + (0.951 + 0.309i)9-s + (0.587 + 0.809i)11-s + (−0.309 − 0.951i)16-s + (−0.412 + 0.809i)17-s + (0.809 − 0.587i)19-s + (−0.951 + 0.309i)20-s + (−1.26 − 1.26i)23-s + (0.309 + 0.951i)25-s + (1.76 + 0.896i)28-s + (0.896 − 1.76i)35-s + (0.809 − 0.587i)36-s + (1.26 − 1.26i)43-s + 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.998 - 0.0527i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.998 - 0.0527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141546430\)
\(L(\frac12)\) \(\approx\) \(1.141546430\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-0.587 + 0.809i)T^{2} \)
3 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \)
23 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (-1.26 + 1.26i)T - iT^{2} \)
47 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.39 - 0.221i)T + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.587 - 0.809i)T^{2} \)
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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

−10.08098680512474593830366414118, −9.233935142769473312081585900198, −8.608141928062725685505363404747, −7.62993732885130493723155809680, −6.72375284973211732524451478424, −5.79750718180171649061335540023, −4.97431489913492046191513615541, −4.16719951670227916784774001988, −2.45028556393840762608289219288, −1.64033749352324593505007525334, 1.30790359288508258780122014736, 3.16636252403556442422088020047, 3.87805617599714412630245613422, 4.36989187883394538345081472708, 6.23292439865907912634744807356, 7.03579202627393213648223197400, 7.60320820713597594464349574950, 7.954664044607568525834951024361, 9.398882470347733718969361366619, 10.28449498101639950449914673728

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