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Properties

Label 2-1045-5.4-c1-0-71
Degree $2$
Conductor $1045$
Sign $-0.980 + 0.195i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.45i·2-s − 0.0791i·3-s − 0.106·4-s + (−2.19 + 0.436i)5-s − 0.114·6-s − 2.96i·7-s − 2.74i·8-s + 2.99·9-s + (0.633 + 3.18i)10-s + 11-s + 0.00845i·12-s + 3.94i·13-s − 4.29·14-s + (0.0345 + 0.173i)15-s − 4.20·16-s − 4.79i·17-s + ⋯
L(s)  = 1  − 1.02i·2-s − 0.0456i·3-s − 0.0534·4-s + (−0.980 + 0.195i)5-s − 0.0468·6-s − 1.11i·7-s − 0.971i·8-s + 0.997·9-s + (0.200 + 1.00i)10-s + 0.301·11-s + 0.00244i·12-s + 1.09i·13-s − 1.14·14-s + (0.00891 + 0.0448i)15-s − 1.05·16-s − 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.980 + 0.195i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.340673852\)
\(L(\frac12)\) \(\approx\) \(1.340673852\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.19 - 0.436i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 + 1.45iT - 2T^{2} \)
3 \( 1 + 0.0791iT - 3T^{2} \)
7 \( 1 + 2.96iT - 7T^{2} \)
13 \( 1 - 3.94iT - 13T^{2} \)
17 \( 1 + 4.79iT - 17T^{2} \)
23 \( 1 + 5.69iT - 23T^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 + 8.56T + 31T^{2} \)
37 \( 1 + 6.70iT - 37T^{2} \)
41 \( 1 - 5.78T + 41T^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 + 3.44iT - 47T^{2} \)
53 \( 1 + 2.73iT - 53T^{2} \)
59 \( 1 - 1.87T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 + 10.1iT - 67T^{2} \)
71 \( 1 + 5.59T + 71T^{2} \)
73 \( 1 + 3.73iT - 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 - 9.90iT - 83T^{2} \)
89 \( 1 - 1.30T + 89T^{2} \)
97 \( 1 + 9.85iT - 97T^{2} \)
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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

−9.697853770441136466366558203549, −9.027928616106544455966095174270, −7.53354409424612934955762900570, −7.20062703481570456995374722163, −6.46897500857165734477738378334, −4.46784229893615632551023821537, −4.18329801880921390246978020337, −3.19839596697690734500877728239, −1.86206970057044089174649898050, −0.61007099138943751245373446268, 1.75212225736311356009815934512, 3.24377130040280697119923070520, 4.32252926581964039853487555193, 5.44790047912838882036833614785, 5.98058120568882940214989596590, 7.19014126503807292438061839425, 7.62915617613907063874440842413, 8.507269183307884453130015060936, 9.084280803563677209514315509270, 10.32539672013676988663790630040

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