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Properties

Label 1-1045-1045.943-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: $\chi_{1045} (943, \cdot )$
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

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

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