Properties

Label 2-1045-209.208-c1-0-30
Degree $2$
Conductor $1045$
Sign $0.216 - 0.976i$
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  − 2.69·2-s + 1.09i·3-s + 5.23·4-s + 5-s − 2.93i·6-s + 4.34i·7-s − 8.70·8-s + 1.80·9-s − 2.69·10-s + (−1.85 − 2.75i)11-s + 5.71i·12-s + 5.00·13-s − 11.6i·14-s + 1.09i·15-s + 12.9·16-s − 1.42i·17-s + ⋯
L(s)  = 1  − 1.90·2-s + 0.630i·3-s + 2.61·4-s + 0.447·5-s − 1.19i·6-s + 1.64i·7-s − 3.07·8-s + 0.602·9-s − 0.850·10-s + (−0.558 − 0.829i)11-s + 1.65i·12-s + 1.38·13-s − 3.12i·14-s + 0.281i·15-s + 3.23·16-s − 0.346i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.216 - 0.976i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.216 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8530793713\)
\(L(\frac12)\) \(\approx\) \(0.8530793713\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 + (1.85 + 2.75i)T \)
19 \( 1 + (-3.00 - 3.15i)T \)
good2 \( 1 + 2.69T + 2T^{2} \)
3 \( 1 - 1.09iT - 3T^{2} \)
7 \( 1 - 4.34iT - 7T^{2} \)
13 \( 1 - 5.00T + 13T^{2} \)
17 \( 1 + 1.42iT - 17T^{2} \)
23 \( 1 - 2.49T + 23T^{2} \)
29 \( 1 - 5.84T + 29T^{2} \)
31 \( 1 + 2.48iT - 31T^{2} \)
37 \( 1 - 6.42iT - 37T^{2} \)
41 \( 1 - 5.23T + 41T^{2} \)
43 \( 1 + 12.1iT - 43T^{2} \)
47 \( 1 - 3.25T + 47T^{2} \)
53 \( 1 + 4.67iT - 53T^{2} \)
59 \( 1 + 1.28iT - 59T^{2} \)
61 \( 1 + 12.8iT - 61T^{2} \)
67 \( 1 + 5.61iT - 67T^{2} \)
71 \( 1 - 10.8iT - 71T^{2} \)
73 \( 1 - 8.64iT - 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 - 12.0iT - 83T^{2} \)
89 \( 1 + 5.26iT - 89T^{2} \)
97 \( 1 + 1.44iT - 97T^{2} \)
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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

−9.878564900876336593133041038030, −9.272372584244014088045786708973, −8.577225502584229647852510675340, −8.140124526925267469948188678179, −6.88139227994217100172680876017, −5.99919194031314030712393251510, −5.34946679058779654733142663263, −3.37201990379563216636691754328, −2.42028547289209123356808033546, −1.20410331567771446252533983101, 0.919225044208578090288791849676, 1.49285946568638465511338286023, 2.86509188824979290593008368164, 4.36150332231328195857984078109, 6.07506110330580370460541149707, 6.88356852524767808746258844818, 7.37338555586358582304325706596, 7.962025867036575845746234521391, 8.973953000073148432957834449438, 9.753207340178353200586718868599

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