Properties

Label 2-1045-5.4-c1-0-77
Degree $2$
Conductor $1045$
Sign $-0.686 + 0.727i$
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  + 0.202i·2-s − 2.47i·3-s + 1.95·4-s + (−1.53 + 1.62i)5-s + 0.501·6-s − 5.06i·7-s + 0.801i·8-s − 3.14·9-s + (−0.329 − 0.310i)10-s + 11-s − 4.85i·12-s − 1.72i·13-s + 1.02·14-s + (4.03 + 3.80i)15-s + 3.75·16-s − 2.65i·17-s + ⋯
L(s)  = 1  + 0.143i·2-s − 1.43i·3-s + 0.979·4-s + (−0.686 + 0.727i)5-s + 0.204·6-s − 1.91i·7-s + 0.283i·8-s − 1.04·9-s + (−0.104 − 0.0981i)10-s + 0.301·11-s − 1.40i·12-s − 0.479i·13-s + 0.273·14-s + (1.04 + 0.981i)15-s + 0.938·16-s − 0.642i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.686 + 0.727i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.604757262\)
\(L(\frac12)\) \(\approx\) \(1.604757262\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.53 - 1.62i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - 0.202iT - 2T^{2} \)
3 \( 1 + 2.47iT - 3T^{2} \)
7 \( 1 + 5.06iT - 7T^{2} \)
13 \( 1 + 1.72iT - 13T^{2} \)
17 \( 1 + 2.65iT - 17T^{2} \)
23 \( 1 - 6.75iT - 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 - 6.13T + 31T^{2} \)
37 \( 1 + 7.00iT - 37T^{2} \)
41 \( 1 + 3.72T + 41T^{2} \)
43 \( 1 - 0.935iT - 43T^{2} \)
47 \( 1 + 3.29iT - 47T^{2} \)
53 \( 1 + 0.668iT - 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 - 12.8iT - 67T^{2} \)
71 \( 1 - 0.833T + 71T^{2} \)
73 \( 1 + 1.38iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 - 1.51T + 89T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−9.879218070460919163457063908932, −8.207073786269685713711406979818, −7.49333798009110600546014800719, −7.21197346354575067509900183693, −6.72111613330234854754805566848, −5.68700171006204707959109443822, −4.01087832065677063325517054533, −3.16994759224339543300214230800, −1.87642269890350815160960608969, −0.70106495450076700557380054277, 1.93649284272224245237408260148, 3.06682071482610233421442281895, 4.04845107906878561378842977443, 4.98282003835786038685711029197, 5.78403965634971507713796492869, 6.66532188061995988587591016773, 8.128396514915478558742237002868, 8.684470805051194297069843405404, 9.395669816131151207828671635627, 10.16196797446133005692512156118

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