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Properties

Label 2-1045-209.208-c1-0-10
Degree $2$
Conductor $1045$
Sign $-0.344 - 0.938i$
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  − 1.10·2-s − 0.279i·3-s − 0.775·4-s + 5-s + 0.308i·6-s + 2.99i·7-s + 3.07·8-s + 2.92·9-s − 1.10·10-s + (−3.17 + 0.950i)11-s + 0.216i·12-s − 0.947·13-s − 3.31i·14-s − 0.279i·15-s − 1.84·16-s + 0.958i·17-s + ⋯
L(s)  = 1  − 0.782·2-s − 0.161i·3-s − 0.387·4-s + 0.447·5-s + 0.126i·6-s + 1.13i·7-s + 1.08·8-s + 0.974·9-s − 0.349·10-s + (−0.958 + 0.286i)11-s + 0.0625i·12-s − 0.262·13-s − 0.885i·14-s − 0.0720i·15-s − 0.461·16-s + 0.232i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.344 - 0.938i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.344 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7040519799\)
\(L(\frac12)\) \(\approx\) \(0.7040519799\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.17 - 0.950i)T \)
19 \( 1 + (-0.265 - 4.35i)T \)
good2 \( 1 + 1.10T + 2T^{2} \)
3 \( 1 + 0.279iT - 3T^{2} \)
7 \( 1 - 2.99iT - 7T^{2} \)
13 \( 1 + 0.947T + 13T^{2} \)
17 \( 1 - 0.958iT - 17T^{2} \)
23 \( 1 + 4.10T + 23T^{2} \)
29 \( 1 - 5.40T + 29T^{2} \)
31 \( 1 + 1.45iT - 31T^{2} \)
37 \( 1 + 9.06iT - 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 8.70iT - 43T^{2} \)
47 \( 1 + 6.30T + 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 + 11.8iT - 59T^{2} \)
61 \( 1 - 3.81iT - 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 - 16.1iT - 71T^{2} \)
73 \( 1 - 8.49iT - 73T^{2} \)
79 \( 1 + 9.49T + 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 + 4.52iT - 89T^{2} \)
97 \( 1 - 12.4iT - 97T^{2} \)
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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

−9.989055931970737312471499220539, −9.497618186250727098803945430019, −8.449690053888247844784794355150, −7.971859339536764684115123152481, −7.01937034170551245155564691116, −5.87483764125146087157570586224, −5.07990166208368113297635343693, −4.08734729273797303066749636133, −2.48613738811042135221274861738, −1.50534634833814020848605863239, 0.43823945159621229731577651727, 1.74437914828672241136801495938, 3.35213331896838168188768154811, 4.58601296178622407117082367283, 5.02437255489830229122393601014, 6.59038836304346715201286224609, 7.31485101166715855774685248233, 8.064566905345294439282777446858, 8.891246951591447981507846310566, 9.961384347072679211781273301372

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