Properties

Label 2-1045-1045.227-c0-0-0
Degree $2$
Conductor $1045$
Sign $0.885 - 0.465i$
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  + (0.951 − 0.309i)4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.412i)7-s + (−0.587 + 0.809i)9-s + (0.951 + 0.309i)11-s + (0.809 − 0.587i)16-s + (−0.0489 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.587 + 0.809i)20-s + (1.39 − 1.39i)23-s + (−0.809 + 0.587i)25-s + (−0.896 − 0.142i)28-s + (0.142 − 0.896i)35-s + (−0.309 + 0.951i)36-s + (−1.39 − 1.39i)43-s + 0.999·44-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.412i)7-s + (−0.587 + 0.809i)9-s + (0.951 + 0.309i)11-s + (0.809 − 0.587i)16-s + (−0.0489 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.587 + 0.809i)20-s + (1.39 − 1.39i)23-s + (−0.809 + 0.587i)25-s + (−0.896 − 0.142i)28-s + (0.142 − 0.896i)35-s + (−0.309 + 0.951i)36-s + (−1.39 − 1.39i)43-s + 0.999·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.885 - 0.465i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.885 - 0.465i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.240136574\)
\(L(\frac12)\) \(\approx\) \(1.240136574\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (-0.951 + 0.309i)T^{2} \)
3 \( 1 + (0.587 - 0.809i)T^{2} \)
7 \( 1 + (0.809 + 0.412i)T + (0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \)
23 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
47 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.951 - 0.309i)T^{2} \)
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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

−10.40299027851685110339392460519, −9.616202831958590247556163256259, −8.486796404254573692869262747553, −7.43601106532249989247807844853, −6.62384146548403989371945491649, −6.31836935589035941410877751032, −5.18572136423179205982101265104, −3.70503937752948131578165567643, −2.81605443740388489398273946657, −1.81024450389923655527135178909, 1.32945827878924088114668928209, 2.82967988266207884503276162076, 3.56061288568117692154673103124, 4.95701527880539416270085085726, 6.05129504686815895134734286029, 6.49013462795956675808768915428, 7.46216259221192162768545942451, 8.686988729004464045132040470627, 9.132504659882844439741009379474, 9.808288829702025144776560349954

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