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Properties

Label 2-1045-209.208-c1-0-14
Degree $2$
Conductor $1045$
Sign $-0.569 - 0.822i$
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  − 2.34·2-s + 1.68i·3-s + 3.51·4-s + 5-s − 3.96i·6-s − 0.129i·7-s − 3.56·8-s + 0.156·9-s − 2.34·10-s + (2.08 + 2.57i)11-s + 5.92i·12-s − 0.796·13-s + 0.303i·14-s + 1.68i·15-s + 1.33·16-s + 0.145i·17-s + ⋯
L(s)  = 1  − 1.66·2-s + 0.973i·3-s + 1.75·4-s + 0.447·5-s − 1.61i·6-s − 0.0487i·7-s − 1.25·8-s + 0.0521·9-s − 0.742·10-s + (0.629 + 0.777i)11-s + 1.71i·12-s − 0.220·13-s + 0.0809i·14-s + 0.435i·15-s + 0.333·16-s + 0.0352i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.569 - 0.822i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6897068834\)
\(L(\frac12)\) \(\approx\) \(0.6897068834\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.08 - 2.57i)T \)
19 \( 1 + (4.34 + 0.326i)T \)
good2 \( 1 + 2.34T + 2T^{2} \)
3 \( 1 - 1.68iT - 3T^{2} \)
7 \( 1 + 0.129iT - 7T^{2} \)
13 \( 1 + 0.796T + 13T^{2} \)
17 \( 1 - 0.145iT - 17T^{2} \)
23 \( 1 + 5.91T + 23T^{2} \)
29 \( 1 - 9.53T + 29T^{2} \)
31 \( 1 - 1.20iT - 31T^{2} \)
37 \( 1 - 10.6iT - 37T^{2} \)
41 \( 1 - 2.99T + 41T^{2} \)
43 \( 1 - 5.01iT - 43T^{2} \)
47 \( 1 + 4.18T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + 14.0iT - 59T^{2} \)
61 \( 1 - 8.51iT - 61T^{2} \)
67 \( 1 - 7.38iT - 67T^{2} \)
71 \( 1 - 8.32iT - 71T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 - 2.27T + 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 - 6.38iT - 89T^{2} \)
97 \( 1 - 0.618iT - 97T^{2} \)
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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

−9.980738347670566630773379643649, −9.650715364612132898964665385131, −8.688305304497635204577176424929, −8.103270994159829915756034035744, −6.91680954643199591788715828049, −6.39639668748371938639782719115, −4.90347529951787260219262915135, −4.06728138393774995994964525531, −2.51553900435238015611273606696, −1.38374582415792026424729958144, 0.57510676343252214229687829143, 1.68738419866989411130984764658, 2.53199684946693444560784058023, 4.23700030774739880799571236730, 5.98107395252436900163337236979, 6.48542085185311845261204120429, 7.34764667724918578659980644299, 8.065060224505146335363997013528, 8.776239623246332018566505175683, 9.462844734410741957971655643443

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