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Properties

Label 1-1045-1045.118-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.742 - 0.669i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.529 + 0.848i)2-s + (0.829 + 0.559i)3-s + (−0.438 + 0.898i)4-s + (−0.0348 + 0.999i)6-s + (−0.994 − 0.104i)7-s + (−0.994 + 0.104i)8-s + (0.374 + 0.927i)9-s + (−0.866 + 0.5i)12-s + (0.788 + 0.615i)13-s + (−0.438 − 0.898i)14-s + (−0.615 − 0.788i)16-s + (−0.927 − 0.374i)17-s + (−0.587 + 0.809i)18-s + (−0.766 − 0.642i)21-s + (−0.984 + 0.173i)23-s + (−0.882 − 0.469i)24-s + ⋯
L(s)  = 1  + (0.529 + 0.848i)2-s + (0.829 + 0.559i)3-s + (−0.438 + 0.898i)4-s + (−0.0348 + 0.999i)6-s + (−0.994 − 0.104i)7-s + (−0.994 + 0.104i)8-s + (0.374 + 0.927i)9-s + (−0.866 + 0.5i)12-s + (0.788 + 0.615i)13-s + (−0.438 − 0.898i)14-s + (−0.615 − 0.788i)16-s + (−0.927 − 0.374i)17-s + (−0.587 + 0.809i)18-s + (−0.766 − 0.642i)21-s + (−0.984 + 0.173i)23-s + (−0.882 − 0.469i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.742 - 0.669i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.742 - 0.669i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4623598776 + 1.203436507i\)
\(L(\frac12)\) \(\approx\) \(-0.4623598776 + 1.203436507i\)
\(L(1)\) \(\approx\) \(0.8060859075 + 0.9740480234i\)
\(L(1)\) \(\approx\) \(0.8060859075 + 0.9740480234i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.529 + 0.848i)T \)
3 \( 1 + (0.829 + 0.559i)T \)
7 \( 1 + (-0.994 - 0.104i)T \)
13 \( 1 + (0.788 + 0.615i)T \)
17 \( 1 + (-0.927 - 0.374i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.997 + 0.0697i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.559 + 0.829i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.970 + 0.241i)T \)
53 \( 1 + (-0.139 + 0.990i)T \)
59 \( 1 + (0.241 + 0.970i)T \)
61 \( 1 + (0.882 - 0.469i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (0.990 - 0.139i)T \)
73 \( 1 + (0.275 - 0.961i)T \)
79 \( 1 + (0.0348 + 0.999i)T \)
83 \( 1 + (-0.743 - 0.669i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.529 - 0.848i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.825470113114860336193422270386, −20.19742943748579818458612889894, −19.72293502283398724998494534241, −18.85908599455563047336894868155, −18.3544301049458680010188418002, −17.49486409270591340808677398517, −16.02469632686673504282214820542, −15.3855852752265151192295190377, −14.578772260612615401816147881490, −13.67720543023131890014069887295, −13.07456999572363994812767023228, −12.63539989486882335299746066411, −11.676477563310242295740746395, −10.66261951155598639777623371636, −9.856445974200080598093264133593, −9.03105718520874824256809295576, −8.381457456214925628091512512907, −7.07789979681822213377957280287, −6.2538525793339445967155422468, −5.45962977763100060311209879189, −3.83371369049725827913729623234, −3.64194802064998474950545071463, −2.45988924132560384522700475928, −1.77217289533957340906267984066, −0.368087576450314148368736088329, 2.009300968031747144106019084032, 3.16894863583427396787572180838, 3.791801779437890607372166554850, 4.56140152760788243993197937821, 5.6570175374641511915724190927, 6.590339070442056788279398801163, 7.31816729339571685181379602164, 8.34017135195034435965353722852, 9.06697276420081506684219739257, 9.65603769047697674657545725655, 10.78671908196632251120233619935, 11.87075487199106105382631517976, 12.96858227592325977075194840628, 13.54790493607429317389523122481, 14.07559082459467904241405095571, 15.133433908619187331999806491813, 15.65870607250867231608499437154, 16.35749845739814243923404476568, 16.89012703040321768920694497777, 18.212679838197518127086199469388, 18.80068293300909880105176582712, 19.95297020008471143095465528162, 20.45329608496169285964914811945, 21.467485151096047923218709405906, 22.08982580015619651924390983366

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