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Properties

Label 1-1805-1805.1047-r0-0-0
Degree $1$
Conductor $1805$
Sign $-0.0497 + 0.998i$
Analytic cond. $8.38238$
Root an. cond. $8.38238$
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  + (−0.908 − 0.418i)2-s + (0.948 + 0.315i)3-s + (0.649 + 0.760i)4-s + (−0.729 − 0.684i)6-s + (0.936 + 0.350i)7-s + (−0.272 − 0.962i)8-s + (0.800 + 0.599i)9-s + (−0.998 − 0.0550i)11-s + (0.376 + 0.926i)12-s + (−0.783 − 0.621i)13-s + (−0.703 − 0.710i)14-s + (−0.155 + 0.987i)16-s + (−0.182 + 0.983i)17-s + (−0.475 − 0.879i)18-s + (0.777 + 0.628i)21-s + (0.883 + 0.467i)22-s + ⋯
L(s)  = 1  + (−0.908 − 0.418i)2-s + (0.948 + 0.315i)3-s + (0.649 + 0.760i)4-s + (−0.729 − 0.684i)6-s + (0.936 + 0.350i)7-s + (−0.272 − 0.962i)8-s + (0.800 + 0.599i)9-s + (−0.998 − 0.0550i)11-s + (0.376 + 0.926i)12-s + (−0.783 − 0.621i)13-s + (−0.703 − 0.710i)14-s + (−0.155 + 0.987i)16-s + (−0.182 + 0.983i)17-s + (−0.475 − 0.879i)18-s + (0.777 + 0.628i)21-s + (0.883 + 0.467i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0497 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0497 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.0497 + 0.998i$
Analytic conductor: \(8.38238\)
Root analytic conductor: \(8.38238\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1805,\ (0:\ ),\ -0.0497 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8005518517 + 0.8414442672i\)
\(L(\frac12)\) \(\approx\) \(0.8005518517 + 0.8414442672i\)
\(L(1)\) \(\approx\) \(0.9231437324 + 0.1530634394i\)
\(L(1)\) \(\approx\) \(0.9231437324 + 0.1530634394i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.908 - 0.418i)T \)
3 \( 1 + (0.948 + 0.315i)T \)
7 \( 1 + (0.936 + 0.350i)T \)
11 \( 1 + (-0.998 - 0.0550i)T \)
13 \( 1 + (-0.783 - 0.621i)T \)
17 \( 1 + (-0.182 + 0.983i)T \)
23 \( 1 + (-0.200 + 0.979i)T \)
29 \( 1 + (0.870 - 0.492i)T \)
31 \( 1 + (-0.904 - 0.426i)T \)
37 \( 1 + (-0.837 - 0.546i)T \)
41 \( 1 + (-0.606 + 0.794i)T \)
43 \( 1 + (-0.443 + 0.896i)T \)
47 \( 1 + (-0.973 - 0.227i)T \)
53 \( 1 + (0.289 + 0.957i)T \)
59 \( 1 + (0.384 + 0.922i)T \)
61 \( 1 + (-0.435 + 0.900i)T \)
67 \( 1 + (0.994 - 0.100i)T \)
71 \( 1 + (0.435 + 0.900i)T \)
73 \( 1 + (-0.942 + 0.333i)T \)
79 \( 1 + (0.832 - 0.554i)T \)
83 \( 1 + (0.981 + 0.191i)T \)
89 \( 1 + (-0.333 + 0.942i)T \)
97 \( 1 + (-0.994 - 0.100i)T \)
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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

−19.96071783243005692296143636477, −19.16277890193570528342299792205, −18.39705618846903735315040441287, −18.04014970707129013173380455327, −17.18463710863104098600602160136, −16.2812082981871103133821016873, −15.61522116326834866986903683433, −14.79755456747670985593970066988, −14.21674817830457201043417775794, −13.67443260401402792420215752177, −12.47660373178264305193633270046, −11.718623552553553614575028228160, −10.70476106783582527573744072746, −10.09703221504725718393587639485, −9.26130152018408496139896536527, −8.44620188418439568595509497822, −7.99376739879620800019678774879, −7.06492008296386019527896911470, −6.778792604080917428992361227454, −5.189315202614082433465736090812, −4.72494687574326481226207763341, −3.27860683461851652723934605896, −2.24799904418132767375787413116, −1.776125079435300278374369891594, −0.45536318107749714949762953834, 1.3670257512436232326882994272, 2.17887408853666616579971321037, 2.8261010736636241824975228582, 3.75892697883280249440235139011, 4.75413133745213177865581061890, 5.68984562901688938024732078259, 7.070773263875561356152631541446, 7.92193836342696961820404886720, 8.1391683071891233930655625494, 8.9963859621332834637383555251, 9.87690525681639563826323515402, 10.42970675824727860399807389469, 11.15552666754528996457212799520, 12.1078570113887890656369354844, 12.90794401875004782215157115053, 13.5901434992311541817452490471, 14.78134573674685216927360047897, 15.1987018914882785959427276597, 15.88940327467293680592870793186, 16.799965241224687327477523464852, 17.726368918213041261402414361271, 18.13390544779787827090428148729, 19.04251025561888011091918257995, 19.66883641187058518126264719228, 20.248176353327037441882924080382

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