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Properties

Label 1-1045-1045.344-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.991 + 0.127i$
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  + (−0.615 + 0.788i)2-s + (−0.719 − 0.694i)3-s + (−0.241 − 0.970i)4-s + (0.990 − 0.139i)6-s + (−0.913 + 0.406i)7-s + (0.913 + 0.406i)8-s + (0.0348 + 0.999i)9-s + (−0.5 + 0.866i)12-s + (−0.882 − 0.469i)13-s + (0.241 − 0.970i)14-s + (−0.882 + 0.469i)16-s + (−0.0348 + 0.999i)17-s + (−0.809 − 0.587i)18-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.374 − 0.927i)24-s + ⋯
L(s)  = 1  + (−0.615 + 0.788i)2-s + (−0.719 − 0.694i)3-s + (−0.241 − 0.970i)4-s + (0.990 − 0.139i)6-s + (−0.913 + 0.406i)7-s + (0.913 + 0.406i)8-s + (0.0348 + 0.999i)9-s + (−0.5 + 0.866i)12-s + (−0.882 − 0.469i)13-s + (0.241 − 0.970i)14-s + (−0.882 + 0.469i)16-s + (−0.0348 + 0.999i)17-s + (−0.809 − 0.587i)18-s + (0.939 + 0.342i)21-s + (−0.766 − 0.642i)23-s + (−0.374 − 0.927i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2305152884 + 0.01472589237i\)
\(L(\frac12)\) \(\approx\) \(0.2305152884 + 0.01472589237i\)
\(L(1)\) \(\approx\) \(0.4028282049 + 0.06499293371i\)
\(L(1)\) \(\approx\) \(0.4028282049 + 0.06499293371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.615 + 0.788i)T \)
3 \( 1 + (-0.719 - 0.694i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.882 - 0.469i)T \)
17 \( 1 + (-0.0348 + 0.999i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.719 + 0.694i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.559 + 0.829i)T \)
53 \( 1 + (0.848 - 0.529i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (-0.374 + 0.927i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.848 - 0.529i)T \)
73 \( 1 + (-0.438 + 0.898i)T \)
79 \( 1 + (-0.990 - 0.139i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.615 + 0.788i)T \)
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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

−21.4127394202946089777607455799, −20.40203754051404230837234789064, −19.93547899238734563425907486713, −19.01497567404311072895869258739, −18.2079605222302561890641106620, −17.425450738653082956906438835722, −16.636082640382106670494439285391, −16.253252956753306034827644049474, −15.3276091708242272998950699673, −14.08512472666842274760270312964, −13.22725501762088138807226371005, −12.189691663613958112173498022742, −11.82112521157934136544456686223, −10.788811653877930782699413368897, −10.130994867596525049581134969662, −9.47499872571537068302036497788, −8.89415000752196106249945268177, −7.384387215283125036359092063055, −6.88930612122256627795832383554, −5.5858969685042213219884994245, −4.601587874105765686601577413855, −3.72181745877159964682464541879, −2.97896871482449420362515020320, −1.65907878722395121125078602548, −0.27888912749628293984209211249, 0.22525362040206930007465444206, 1.57927539749527187935342765207, 2.54853416266484736764131885075, 4.16459494763612183152370253361, 5.34742893056703488905640847210, 5.96456046630957226414509444716, 6.62911850884809195577655756649, 7.52259441884832364390856747179, 8.18885395478857947130354237533, 9.27642558259133592027553395230, 10.08796561899959078265587966473, 10.796676236828396916358660843557, 11.86813759291984471015492724836, 12.78176483509331998575107596139, 13.293880309142088891377062848340, 14.51231462672505173433691716687, 15.16481281444428303914100692469, 16.23136837773927122193824309878, 16.62843622748510869991238881750, 17.509041527104888507361532845491, 18.09799907185596273076821549868, 18.96808830048360121581656996531, 19.42809041557813803580871881433, 20.183537087266490629749847756, 21.714311233985437697528287978807

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