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Properties

Label 1-1303-1303.1058-r0-0-0
Degree $1$
Conductor $1303$
Sign $0.836 - 0.548i$
Analytic cond. $6.05110$
Root an. cond. $6.05110$
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  + (0.885 − 0.464i)2-s + (0.949 + 0.313i)3-s + (0.569 − 0.822i)4-s + (0.820 + 0.571i)5-s + (0.986 − 0.163i)6-s + (0.792 − 0.610i)7-s + (0.122 − 0.992i)8-s + (0.803 + 0.594i)9-s + (0.992 + 0.125i)10-s + (−0.874 − 0.485i)11-s + (0.798 − 0.602i)12-s + (0.997 + 0.0675i)13-s + (0.418 − 0.908i)14-s + (0.600 + 0.799i)15-s + (−0.351 − 0.936i)16-s + (−0.999 + 0.0144i)17-s + ⋯
L(s)  = 1  + (0.885 − 0.464i)2-s + (0.949 + 0.313i)3-s + (0.569 − 0.822i)4-s + (0.820 + 0.571i)5-s + (0.986 − 0.163i)6-s + (0.792 − 0.610i)7-s + (0.122 − 0.992i)8-s + (0.803 + 0.594i)9-s + (0.992 + 0.125i)10-s + (−0.874 − 0.485i)11-s + (0.798 − 0.602i)12-s + (0.997 + 0.0675i)13-s + (0.418 − 0.908i)14-s + (0.600 + 0.799i)15-s + (−0.351 − 0.936i)16-s + (−0.999 + 0.0144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1303 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1303 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1303\)
Sign: $0.836 - 0.548i$
Analytic conductor: \(6.05110\)
Root analytic conductor: \(6.05110\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1303} (1058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1303,\ (0:\ ),\ 0.836 - 0.548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.541258260 - 1.355447479i\)
\(L(\frac12)\) \(\approx\) \(4.541258260 - 1.355447479i\)
\(L(1)\) \(\approx\) \(2.759077183 - 0.5779091267i\)
\(L(1)\) \(\approx\) \(2.759077183 - 0.5779091267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1303 \( 1 \)
good2 \( 1 + (0.885 - 0.464i)T \)
3 \( 1 + (0.949 + 0.313i)T \)
5 \( 1 + (0.820 + 0.571i)T \)
7 \( 1 + (0.792 - 0.610i)T \)
11 \( 1 + (-0.874 - 0.485i)T \)
13 \( 1 + (0.997 + 0.0675i)T \)
17 \( 1 + (-0.999 + 0.0144i)T \)
19 \( 1 + (0.0843 + 0.996i)T \)
23 \( 1 + (-0.993 + 0.110i)T \)
29 \( 1 + (-0.156 - 0.987i)T \)
31 \( 1 + (0.681 + 0.731i)T \)
37 \( 1 + (0.170 - 0.985i)T \)
41 \( 1 + (-0.995 + 0.0915i)T \)
43 \( 1 + (0.929 - 0.367i)T \)
47 \( 1 + (-0.117 + 0.993i)T \)
53 \( 1 + (-0.369 + 0.929i)T \)
59 \( 1 + (-0.834 - 0.551i)T \)
61 \( 1 + (-0.935 - 0.354i)T \)
67 \( 1 + (0.504 - 0.863i)T \)
71 \( 1 + (0.487 + 0.873i)T \)
73 \( 1 + (-0.938 - 0.345i)T \)
79 \( 1 + (-0.678 + 0.734i)T \)
83 \( 1 + (-0.999 - 0.00482i)T \)
89 \( 1 + (-0.980 + 0.196i)T \)
97 \( 1 + (0.631 - 0.775i)T \)
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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

−21.11700160964102040187331335137, −20.368032053309177248124894286021, −20.06785640607766831411995148968, −18.49825881163218992156641337415, −17.9855999590955283484867993711, −17.36072601428994526033340935834, −16.0988726536253242356550337840, −15.50624079223975527958583064134, −14.92080157331522894722815758619, −13.951766536108028957951040004304, −13.43025104267063166824384027544, −12.93064305585021971167507215742, −12.07792155180413884552591888014, −11.130573482447137127610784341394, −10.03451759403704908662502699163, −8.8160157627107903891514782873, −8.5114030674836668452622472751, −7.63831981773551167888229830574, −6.64348673235030616816561591401, −5.85014817631799518568743371311, −4.87951954394209096556135145771, −4.318753355553110345349605840606, −2.98277456189550056407841076571, −2.25106550465534182254729944872, −1.57127798251260034324571097652, 1.42569344054781992646931261492, 2.10103330734601421079994586445, 2.96698590259909951640574364041, 3.841723052198851692255065014919, 4.55771688886589873219040910265, 5.62469796601227135273294092994, 6.36668417169515920560611950997, 7.47611282941014355431067140452, 8.25092706126735758321016700482, 9.3379340252377788847361117914, 10.30549252860050794939509576672, 10.66458222332509611855755138107, 11.40463448437592138081308783523, 12.7451328196885528253117127737, 13.48522254188861181140471356396, 14.060463461151886681464869673216, 14.282886473924007412045053979310, 15.533427224805050422790261935771, 15.79993382762062267839600723996, 17.06597594275014873157069116006, 18.27333227012391805798058074977, 18.60356434979311156628544716504, 19.66612001469447386561412200292, 20.42388875198939520510089814952, 21.08681397937065067931753406971

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