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Properties

Label 1-1045-1045.169-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.760 + 0.648i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.760 + 0.648i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.760 + 0.648i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6410173179 + 0.2362076744i\)
\(L(\frac12)\) \(\approx\) \(0.6410173179 + 0.2362076744i\)
\(L(1)\) \(\approx\) \(0.5621466609 + 0.4441028329i\)
\(L(1)\) \(\approx\) \(0.5621466609 + 0.4441028329i\)

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

−21.61779376965591932987625458488, −20.40450219761386668776476236819, −19.77910010750422184118968887013, −19.35553706857985591543037289023, −18.33688263806158639628393358712, −17.73234375576703565955035959823, −17.02327619513768651717502953117, −16.42223705008092453934859481095, −14.70735854504998239389462028567, −14.17981413229430056865133259890, −13.25744097444917246414370473276, −12.7151322975607129435263565009, −11.91881997006102484592786498620, −11.19896167864363841664292171411, −10.3511286939348122517071576628, −9.756400449636991168835126710093, −8.41279646964497914732817250997, −7.75003808125054618674431299532, −6.92604584092872226160867663434, −5.71817478882644743950608727600, −4.89204271833761727043770517378, −3.87905547294265211140906631496, −2.85390078547700345893271086401, −1.76014805554389157313311092690, −0.99780747205796760995674434435, 0.378145385406715024990659336145, 2.35440298070680317086540048360, 3.55578783757959159427663692495, 4.58935840216181548107333488335, 5.24857544343405695341374665561, 5.91834574085995405781784514311, 6.82546870656026518458776306609, 7.92359758260469792120173687714, 8.73249540363947901931867694315, 9.65745178885380516428122062670, 9.96672993951844151275940656744, 11.34940831620280584710664924506, 12.106945481117029225086002787241, 12.8958283498720679349128935663, 14.33259556314755127000899346393, 14.58279356271623672820768473159, 15.48320017233212645901292747793, 16.15815168117685196711503066129, 16.70255793078231379093037719810, 17.64715539334190488456053932884, 18.210425920644208090445777684171, 19.08074779482376159964462643572, 20.10674470616149888554758461054, 21.2521850757869919144562804522, 21.73827814671002076420945206853

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