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Properties

Label 1-1045-1045.502-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.931 - 0.363i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (0.743 + 0.669i)2-s + (−0.994 − 0.104i)3-s + (0.104 + 0.994i)4-s + (−0.669 − 0.743i)6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s − i·12-s + (−0.207 + 0.978i)13-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (0.207 + 0.978i)17-s + (0.587 + 0.809i)18-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s + (0.669 − 0.743i)24-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (−0.994 − 0.104i)3-s + (0.104 + 0.994i)4-s + (−0.669 − 0.743i)6-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s − i·12-s + (−0.207 + 0.978i)13-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (0.207 + 0.978i)17-s + (0.587 + 0.809i)18-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s + (0.669 − 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.931 - 0.363i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (502, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.931 - 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4055518525 + 2.152260924i\)
\(L(\frac12)\) \(\approx\) \(-0.4055518525 + 2.152260924i\)
\(L(1)\) \(\approx\) \(0.8861276178 + 0.8807781083i\)
\(L(1)\) \(\approx\) \(0.8861276178 + 0.8807781083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (-0.994 - 0.104i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.207 + 0.978i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.406 + 0.913i)T \)
53 \( 1 + (0.207 - 0.978i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (-0.406 + 0.913i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−20.9613574398301012087217469478, −20.4048574591710523399489777597, −19.51963953467811769836401185128, −18.5281292839483606568803632718, −17.850286812109851181091600344041, −17.08443786008029820416186239446, −16.143344701037158998224727694169, −15.35853229481341467917288554776, −14.50422138948961263199571282144, −13.65566135380783107360328037265, −12.86033517425871199567397409961, −12.13385947215497360007792828569, −11.3344008860327291943318351574, −10.66016238391965766015387449218, −10.15054835373637083196365886988, −9.12156768473878426609612132802, −7.59990294345182051650838514501, −6.90908954147868698375049214268, −5.81985177682187201798381921086, −5.09774565164660625657082126731, −4.445320278109081542121552828750, −3.50020705518465663752352295436, −2.32092918519605856848514682695, −1.00458904346662795948440508962, −0.470718099740270117703048597880, 1.386912934819180925174732912749, 2.449147760065105200067577603634, 3.82559757877347524603620438083, 4.68982204932078538894364417002, 5.38968911799112428918695776457, 6.134255284207476665307648490012, 6.9034438495047546698790246027, 7.786746913294519339810357244992, 8.71058815635848415433842110751, 9.698130590216100986426295780084, 11.14291101290545728236036602832, 11.44865737971644424474098509481, 12.43803192787828055068590609375, 12.88755447874983972033788041723, 13.982769798039360050591398564123, 14.86414924548179677644462536363, 15.4245896154548803594909309705, 16.387557357289447110015172295211, 16.96368077474864742873355607028, 17.66023957373561334586036472982, 18.46809623828872430250338172016, 19.22111780055691461223870459815, 20.630433798080706178772411535261, 21.45808198981123182324086581837, 21.806881761854143691264774883068

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