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Properties

Label 1-5273-5273.1078-r0-0-0
Degree $1$
Conductor $5273$
Sign $0.766 + 0.642i$
Analytic cond. $24.4877$
Root an. cond. $24.4877$
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  + (−0.523 − 0.852i)2-s + (−0.704 + 0.709i)3-s + (−0.452 + 0.891i)4-s + (0.939 + 0.343i)5-s + (0.973 + 0.229i)6-s + (−0.816 + 0.576i)7-s + (0.996 − 0.0809i)8-s + (−0.00715 − 0.999i)9-s + (−0.198 − 0.980i)10-s + (−0.535 − 0.844i)11-s + (−0.313 − 0.949i)12-s + (−0.914 + 0.405i)13-s + (0.919 + 0.394i)14-s + (−0.905 + 0.424i)15-s + (−0.590 − 0.807i)16-s + (0.906 − 0.422i)17-s + ⋯
L(s)  = 1  + (−0.523 − 0.852i)2-s + (−0.704 + 0.709i)3-s + (−0.452 + 0.891i)4-s + (0.939 + 0.343i)5-s + (0.973 + 0.229i)6-s + (−0.816 + 0.576i)7-s + (0.996 − 0.0809i)8-s + (−0.00715 − 0.999i)9-s + (−0.198 − 0.980i)10-s + (−0.535 − 0.844i)11-s + (−0.313 − 0.949i)12-s + (−0.914 + 0.405i)13-s + (0.919 + 0.394i)14-s + (−0.905 + 0.424i)15-s + (−0.590 − 0.807i)16-s + (0.906 − 0.422i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5273\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(24.4877\)
Root analytic conductor: \(24.4877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5273} (1078, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5273,\ (0:\ ),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5580583787 + 0.2030580789i\)
\(L(\frac12)\) \(\approx\) \(0.5580583787 + 0.2030580789i\)
\(L(1)\) \(\approx\) \(0.5769011024 + 0.02340529926i\)
\(L(1)\) \(\approx\) \(0.5769011024 + 0.02340529926i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5273 \( 1 \)
good2 \( 1 + (-0.523 - 0.852i)T \)
3 \( 1 + (-0.704 + 0.709i)T \)
5 \( 1 + (0.939 + 0.343i)T \)
7 \( 1 + (-0.816 + 0.576i)T \)
11 \( 1 + (-0.535 - 0.844i)T \)
13 \( 1 + (-0.914 + 0.405i)T \)
17 \( 1 + (0.906 - 0.422i)T \)
19 \( 1 + (-0.0143 + 0.999i)T \)
23 \( 1 + (-0.785 - 0.618i)T \)
29 \( 1 + (-0.935 - 0.354i)T \)
31 \( 1 + (-0.151 - 0.988i)T \)
37 \( 1 + (-0.411 + 0.911i)T \)
41 \( 1 + (-0.987 + 0.156i)T \)
43 \( 1 + (0.309 - 0.950i)T \)
47 \( 1 + (-0.325 - 0.945i)T \)
53 \( 1 + (-0.385 - 0.922i)T \)
59 \( 1 + (0.590 - 0.807i)T \)
61 \( 1 + (-0.531 - 0.847i)T \)
67 \( 1 + (0.144 + 0.989i)T \)
71 \( 1 + (-0.140 + 0.990i)T \)
73 \( 1 + (0.341 + 0.940i)T \)
79 \( 1 + (0.569 + 0.822i)T \)
83 \( 1 + (0.502 - 0.864i)T \)
89 \( 1 + (-0.984 + 0.175i)T \)
97 \( 1 + (-0.469 - 0.882i)T \)
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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

−17.85288878606235711382439890962, −17.25537917039698326087632948557, −16.67989971565068312422248821180, −16.21083966708835833806634242932, −15.38450542758314399370663235831, −14.56308206980608790287976511269, −13.802469497036831775601398337009, −13.30104145908537849723867535484, −12.635624938463635514801894343805, −12.1893487520454329431726904536, −10.725566548929282322172786848599, −10.50033160187567387438575557045, −9.65432379444195141210137513778, −9.31448310137342831147357364762, −8.14841372837518289838584349482, −7.41619809375842842193452602885, −7.09935162564148325325541944085, −6.26155780925829065821432482718, −5.66153002350533265741903986241, −5.08876759683489183684997239035, −4.43882666931461387656176729418, −3.01773961755417498813951306719, −1.98265356149299570631296649605, −1.370335816876912848408517181793, −0.35076348861067385619053886015, 0.56448139889895470961261685701, 1.84791541233386348662391028737, 2.51906477075093807342146539100, 3.347044932945386148138527815017, 3.83780676838992654653229573104, 5.0842507675631704162105015219, 5.509875434971902042162887880247, 6.28621326710400902661221264128, 7.038856723690545625697758748864, 8.09556419264665473690133914858, 8.85609770606383544968284148010, 9.72072195805950812642553736635, 9.92674417829782761161623947154, 10.37020675615129383122322648447, 11.37005339612079097844850916848, 11.83140037590495248651160673522, 12.54586485701848947073300906177, 13.13846864059994282806603343007, 14.01135455170779109979867157123, 14.623685196677256025273144899748, 15.58986001780244468836668936935, 16.39118933769896963063347814036, 16.81132353290135232684229868107, 17.21741538699635486598341562920, 18.27651565410373502356775717680

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