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Properties

Label 2-1045-5.4-c1-0-25
Degree $2$
Conductor $1045$
Sign $-0.447 - 0.894i$
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  + 0.414i·2-s + 2i·3-s + 1.82·4-s + (−1 − 2i)5-s − 0.828·6-s + 0.828i·7-s + 1.58i·8-s − 9-s + (0.828 − 0.414i)10-s − 11-s + 3.65i·12-s + 6.82i·13-s − 0.343·14-s + (4 − 2i)15-s + 3·16-s − 6.82i·17-s + ⋯
L(s)  = 1  + 0.292i·2-s + 1.15i·3-s + 0.914·4-s + (−0.447 − 0.894i)5-s − 0.338·6-s + 0.313i·7-s + 0.560i·8-s − 0.333·9-s + (0.261 − 0.130i)10-s − 0.301·11-s + 1.05i·12-s + 1.89i·13-s − 0.0917·14-s + (1.03 − 0.516i)15-s + 0.750·16-s − 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.774841408\)
\(L(\frac12)\) \(\approx\) \(1.774841408\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1 + 2i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 - 0.414iT - 2T^{2} \)
3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 - 0.828iT - 7T^{2} \)
13 \( 1 - 6.82iT - 13T^{2} \)
17 \( 1 + 6.82iT - 17T^{2} \)
23 \( 1 - 7.65iT - 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 - 6.48T + 41T^{2} \)
43 \( 1 + 0.828iT - 43T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 1.17iT - 73T^{2} \)
79 \( 1 + 5.65T + 79T^{2} \)
83 \( 1 + 6.48iT - 83T^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 + 1.17iT - 97T^{2} \)
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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

−10.03474539479441114259633067695, −9.267538954758033643928019276761, −8.821828430448022579819585489628, −7.52744177956391028333381594611, −7.02163859801641159292072277368, −5.69057376491909091736527160628, −4.96809708116994034877017462531, −4.18521076800457297353933908422, −3.07076017245275117511126261716, −1.65508754891442004233261627156, 0.804195951379066507540454243103, 2.19218758115201353359208174167, 2.97679914715805688795529702647, 4.01333435495089806689972868207, 5.81829463475314533358373380360, 6.34868305805270532060359226539, 7.28535268829373235749235493001, 7.72076602443930312146967439043, 8.405575609961715802681196123811, 10.18703846579078259120018055193

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