Properties

Label 1-1045-1045.158-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.876 - 0.481i$
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  + (−0.139 − 0.990i)2-s + (0.970 − 0.241i)3-s + (−0.961 + 0.275i)4-s + (−0.374 − 0.927i)6-s + (0.406 − 0.913i)7-s + (0.406 + 0.913i)8-s + (0.882 − 0.469i)9-s + (−0.866 + 0.5i)12-s + (−0.529 + 0.848i)13-s + (−0.961 − 0.275i)14-s + (0.848 − 0.529i)16-s + (−0.469 + 0.882i)17-s + (−0.587 − 0.809i)18-s + (0.173 − 0.984i)21-s + (0.342 − 0.939i)23-s + (0.615 + 0.788i)24-s + ⋯
L(s)  = 1  + (−0.139 − 0.990i)2-s + (0.970 − 0.241i)3-s + (−0.961 + 0.275i)4-s + (−0.374 − 0.927i)6-s + (0.406 − 0.913i)7-s + (0.406 + 0.913i)8-s + (0.882 − 0.469i)9-s + (−0.866 + 0.5i)12-s + (−0.529 + 0.848i)13-s + (−0.961 − 0.275i)14-s + (0.848 − 0.529i)16-s + (−0.469 + 0.882i)17-s + (−0.587 − 0.809i)18-s + (0.173 − 0.984i)21-s + (0.342 − 0.939i)23-s + (0.615 + 0.788i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6250995393 - 2.436678803i\)
\(L(\frac12)\) \(\approx\) \(0.6250995393 - 2.436678803i\)
\(L(1)\) \(\approx\) \(1.028431837 - 0.8426240192i\)
\(L(1)\) \(\approx\) \(1.028431837 - 0.8426240192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.139 - 0.990i)T \)
3 \( 1 + (0.970 - 0.241i)T \)
7 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (-0.529 + 0.848i)T \)
17 \( 1 + (-0.469 + 0.882i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (0.719 + 0.694i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.241 - 0.970i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.898 - 0.438i)T \)
53 \( 1 + (0.999 + 0.0348i)T \)
59 \( 1 + (-0.438 - 0.898i)T \)
61 \( 1 + (-0.615 + 0.788i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (0.0348 + 0.999i)T \)
73 \( 1 + (0.0697 - 0.997i)T \)
79 \( 1 + (0.374 - 0.927i)T \)
83 \( 1 + (-0.207 - 0.978i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.139 - 0.990i)T \)
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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

−21.58876679474091619683285378816, −21.13589855449969696521579059431, −19.80008734879399158174959026732, −19.45728918750397847598495465591, −18.24344660812728878741414839525, −18.00287881017048393249810725172, −16.90286575527167513984309082749, −15.81250586496124014080030110852, −15.464522858761198681197048541797, −14.75112919633842701420541383669, −14.03810573515807600227097331841, −13.24515195493323124559810323524, −12.4539272592279075401614945199, −11.25682928134532345131355125288, −9.98730719238234081470285720064, −9.47413706548723779951745394104, −8.61089251409361921404828632194, −7.98685819001698075643611170726, −7.26709229870490965702626255016, −6.19255068748507303956077702981, −5.10192454367550345043883277487, −4.60888803779735778910076954247, −3.28802523213369417699870101899, −2.42950593474673955127538504800, −1.0743880103934216797858933380, 0.524698541572352747056371589643, 1.59073860915187060897159936157, 2.30733846326581358115451511598, 3.36930562681436472024844040597, 4.22975441494059001771280202821, 4.7911858495782503006042506656, 6.537414089521810002607713465379, 7.400081719966093694450977018198, 8.34765640254390687554890103213, 8.858432652619637654417582971865, 9.94598072582183706314334728212, 10.4427571022487437594904418579, 11.46310443452309486190542833938, 12.337097204366564109374369677733, 13.142239909020641366359585475840, 13.82163328057742978773753164069, 14.43454048439116155333061248746, 15.20363482668368374958076323209, 16.6183515166215647711514278227, 17.24684793956340312998850661824, 18.174175530097450958465294038, 18.921814949187732804818192815684, 19.59786432855923314983557807916, 20.18936357316904173530618266343, 20.8527929041488742620948292542

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