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Properties

Label 1-1045-1045.249-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.922 - 0.385i$
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  + (−0.0348 + 0.999i)2-s + (0.438 − 0.898i)3-s + (−0.997 − 0.0697i)4-s + (0.882 + 0.469i)6-s + (−0.104 − 0.994i)7-s + (0.104 − 0.994i)8-s + (−0.615 − 0.788i)9-s + (−0.5 + 0.866i)12-s + (−0.990 + 0.139i)13-s + (0.997 − 0.0697i)14-s + (0.990 + 0.139i)16-s + (−0.615 + 0.788i)17-s + (0.809 − 0.587i)18-s + (−0.939 − 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.848 − 0.529i)24-s + ⋯
L(s)  = 1  + (−0.0348 + 0.999i)2-s + (0.438 − 0.898i)3-s + (−0.997 − 0.0697i)4-s + (0.882 + 0.469i)6-s + (−0.104 − 0.994i)7-s + (0.104 − 0.994i)8-s + (−0.615 − 0.788i)9-s + (−0.5 + 0.866i)12-s + (−0.990 + 0.139i)13-s + (0.997 − 0.0697i)14-s + (0.990 + 0.139i)16-s + (−0.615 + 0.788i)17-s + (0.809 − 0.587i)18-s + (−0.939 − 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.848 − 0.529i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.922 - 0.385i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.922 - 0.385i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06642250204 - 0.3310792963i\)
\(L(\frac12)\) \(\approx\) \(0.06642250204 - 0.3310792963i\)
\(L(1)\) \(\approx\) \(0.7849108989 + 0.02316655835i\)
\(L(1)\) \(\approx\) \(0.7849108989 + 0.02316655835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.0348 + 0.999i)T \)
3 \( 1 + (0.438 - 0.898i)T \)
7 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (-0.990 + 0.139i)T \)
17 \( 1 + (-0.615 + 0.788i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.559 - 0.829i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.438 - 0.898i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.961 - 0.275i)T \)
53 \( 1 + (-0.374 + 0.927i)T \)
59 \( 1 + (-0.961 + 0.275i)T \)
61 \( 1 + (-0.848 + 0.529i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.374 + 0.927i)T \)
73 \( 1 + (-0.719 - 0.694i)T \)
79 \( 1 + (-0.882 + 0.469i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (0.0348 - 0.999i)T \)
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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

−21.60439473234425985807054904525, −21.297299349453667301175159928863, −20.28495993080351887225904661883, −19.66621847207846633107011971370, −19.10192816405400792464946005868, −18.02615349890115129027228383527, −17.45729644553661648517100457431, −16.27766480204180168182475885726, −15.58199070760229951297754974224, −14.66031961524094644644278711378, −14.06992297805079846634629821909, −13.11225605028663511257911774074, −12.17484429194336417927602633586, −11.54070903104755403118940993069, −10.66797718690991432456609767635, −9.72407661819040846633188331067, −9.34267885592622269630469319879, −8.49369785253462650836484650942, −7.65882274136037885677363034585, −6.05132476451131263527178690570, −4.990551624221827807994581790184, −4.5661965618601099367409664218, −3.268980501709432533428021356543, −2.70984031935828503993324961963, −1.83060302495119656459526705328, 0.13328749291111143678372970056, 1.383147300436269604448428003833, 2.67543757237865513881355229334, 3.921912786866868749871426155403, 4.63714396430571734638972291890, 5.96970410682924611050763109971, 6.6458840834463010403274644907, 7.3481405036936924472961358214, 8.0477957464749920901615090356, 8.789254856815597515682655832354, 9.80687910209745902404911681636, 10.57110234704614001855188326916, 12.03682688088837906508444691334, 12.6617584217900416518577215105, 13.67269112121244274124537492893, 13.95615572502912530856089952256, 14.85172333247581308451072981984, 15.61148592192465077780803081798, 16.69592245029735117839338274136, 17.39076189937506034902657656709, 17.769803544211347054273387794165, 18.94066361120476264526251977146, 19.42566698851643308750570008696, 20.17654525746786978265795538940, 21.21752706202047682639536406005

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