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Properties

Label 1-1045-1045.233-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.859 + 0.511i$
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  + (−0.0697 − 0.997i)2-s + (−0.788 − 0.615i)3-s + (−0.990 + 0.139i)4-s + (−0.559 + 0.829i)6-s + (0.207 − 0.978i)7-s + (0.207 + 0.978i)8-s + (0.241 + 0.970i)9-s + (0.866 + 0.5i)12-s + (0.275 − 0.961i)13-s + (−0.990 − 0.139i)14-s + (0.961 − 0.275i)16-s + (−0.970 − 0.241i)17-s + (0.951 − 0.309i)18-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (0.438 − 0.898i)24-s + ⋯
L(s)  = 1  + (−0.0697 − 0.997i)2-s + (−0.788 − 0.615i)3-s + (−0.990 + 0.139i)4-s + (−0.559 + 0.829i)6-s + (0.207 − 0.978i)7-s + (0.207 + 0.978i)8-s + (0.241 + 0.970i)9-s + (0.866 + 0.5i)12-s + (0.275 − 0.961i)13-s + (−0.990 − 0.139i)14-s + (0.961 − 0.275i)16-s + (−0.970 − 0.241i)17-s + (0.951 − 0.309i)18-s + (−0.766 + 0.642i)21-s + (0.984 + 0.173i)23-s + (0.438 − 0.898i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.859 + 0.511i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1992588150 - 0.7247835636i\)
\(L(\frac12)\) \(\approx\) \(-0.1992588150 - 0.7247835636i\)
\(L(1)\) \(\approx\) \(0.4451708395 - 0.5619453493i\)
\(L(1)\) \(\approx\) \(0.4451708395 - 0.5619453493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.0697 - 0.997i)T \)
3 \( 1 + (-0.788 - 0.615i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (0.275 - 0.961i)T \)
17 \( 1 + (-0.970 - 0.241i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (-0.374 + 0.927i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.615 - 0.788i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.529 - 0.848i)T \)
53 \( 1 + (-0.694 - 0.719i)T \)
59 \( 1 + (-0.848 - 0.529i)T \)
61 \( 1 + (-0.438 - 0.898i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.719 - 0.694i)T \)
73 \( 1 + (-0.999 - 0.0348i)T \)
79 \( 1 + (0.559 + 0.829i)T \)
83 \( 1 + (-0.994 + 0.104i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.0697 + 0.997i)T \)
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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

−22.05460096585377408120435923430, −21.510128603234067877045208539281, −20.788312018337977520350012519199, −19.31275642369618766599853553575, −18.672273172580182668647082400873, −17.823613212000615611848854437121, −17.29281108648450134489039546057, −16.407538067922975598153218489281, −15.78996908643607360891981883835, −15.13069087527715570281134936490, −14.49666774162107689353504476904, −13.37251136350444168289008255862, −12.54149080117859895679810019066, −11.58286693237154296830108121935, −10.8861444589159156743091330650, −9.67943523078772063040878740794, −9.13243135870483256807593528503, −8.39839155917625280155924380573, −7.19395961994256236774447279912, −6.213247885983488177449993886573, −5.88392542000472382256456456762, −4.58415411581210308774453379509, −4.40066035806492832913210038800, −2.87497312673054969497388366261, −1.26717390779082264143992045713, 0.4379047128307402733564370852, 1.26575939409746255321853106760, 2.34867459294236291474803339286, 3.464942181484468828557471012466, 4.540945317887585321623706511437, 5.20340404802464681622535451143, 6.32134127371065813113489460941, 7.37625099241590168741586401480, 8.058322789052667709295807922971, 9.14825785644141404456672086360, 10.21060781580589233867231374797, 10.960750578045501230399426958273, 11.24740727588390423055277668968, 12.39554936110636359364953762776, 13.07261261209795038665762137610, 13.55741481843951514044648871211, 14.4886213431950889086524643922, 15.71894817968034786448729445398, 16.79690506686434374094128932667, 17.441365635624117816990276568053, 17.94805400574721739236579356687, 18.75417897671908661686080542726, 19.61272865918042656492115565658, 20.21461278981386758172228409588, 20.98973729737351157695141713366

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