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Properties

Label 1-5245-5245.1513-r1-0-0
Degree $1$
Conductor $5245$
Sign $-0.957 - 0.287i$
Analytic cond. $563.653$
Root an. cond. $563.653$
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  + (0.0718 + 0.997i)2-s + (0.991 − 0.131i)3-s + (−0.989 + 0.143i)4-s + (0.202 + 0.979i)6-s + (−0.603 − 0.797i)7-s + (−0.214 − 0.976i)8-s + (0.965 − 0.260i)9-s + (−0.155 − 0.987i)11-s + (−0.962 + 0.272i)12-s + (−0.374 + 0.927i)13-s + (0.752 − 0.658i)14-s + (0.958 − 0.283i)16-s + (−0.775 − 0.631i)17-s + (0.329 + 0.944i)18-s + (−0.363 + 0.931i)19-s + ⋯
L(s)  = 1  + (0.0718 + 0.997i)2-s + (0.991 − 0.131i)3-s + (−0.989 + 0.143i)4-s + (0.202 + 0.979i)6-s + (−0.603 − 0.797i)7-s + (−0.214 − 0.976i)8-s + (0.965 − 0.260i)9-s + (−0.155 − 0.987i)11-s + (−0.962 + 0.272i)12-s + (−0.374 + 0.927i)13-s + (0.752 − 0.658i)14-s + (0.958 − 0.283i)16-s + (−0.775 − 0.631i)17-s + (0.329 + 0.944i)18-s + (−0.363 + 0.931i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5245 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5245 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5245\)    =    \(5 \cdot 1049\)
Sign: $-0.957 - 0.287i$
Analytic conductor: \(563.653\)
Root analytic conductor: \(563.653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5245} (1513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5245,\ (1:\ ),\ -0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03070136496 - 0.2091619899i\)
\(L(\frac12)\) \(\approx\) \(0.03070136496 - 0.2091619899i\)
\(L(1)\) \(\approx\) \(1.068827953 + 0.2445431076i\)
\(L(1)\) \(\approx\) \(1.068827953 + 0.2445431076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1049 \( 1 \)
good2 \( 1 + (0.0718 + 0.997i)T \)
3 \( 1 + (0.991 - 0.131i)T \)
7 \( 1 + (-0.603 - 0.797i)T \)
11 \( 1 + (-0.155 - 0.987i)T \)
13 \( 1 + (-0.374 + 0.927i)T \)
17 \( 1 + (-0.775 - 0.631i)T \)
19 \( 1 + (-0.363 + 0.931i)T \)
23 \( 1 + (-0.0957 - 0.995i)T \)
29 \( 1 + (-0.998 - 0.0479i)T \)
31 \( 1 + (0.825 + 0.564i)T \)
37 \( 1 + (0.922 - 0.385i)T \)
41 \( 1 + (-0.363 + 0.931i)T \)
43 \( 1 + (0.857 - 0.513i)T \)
47 \( 1 + (-0.260 - 0.965i)T \)
53 \( 1 + (-0.940 - 0.340i)T \)
59 \( 1 + (-0.752 - 0.658i)T \)
61 \( 1 + (0.574 - 0.818i)T \)
67 \( 1 + (-0.190 + 0.981i)T \)
71 \( 1 + (0.202 - 0.979i)T \)
73 \( 1 + (0.948 + 0.318i)T \)
79 \( 1 + (-0.574 - 0.818i)T \)
83 \( 1 + (0.583 - 0.811i)T \)
89 \( 1 + (-0.797 - 0.603i)T \)
97 \( 1 + (0.983 - 0.178i)T \)
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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

−18.24757147762104878144028359836, −17.67145155818783665707441449830, −17.02488372457042671868520453894, −15.695990937408237486030436802559, −15.25732515584835266763910503108, −14.93624167018865599407745832932, −13.95634219857970315640440691194, −13.16731414918640224187545440749, −12.85941942338503662231641608939, −12.33146700415516775653205998954, −11.36714833418264877992988696599, −10.648392925742355104475512836802, −9.85725013311524686483304729113, −9.443669485040668156914345265949, −8.911921191207590160600443437076, −8.06502588112124315937468945247, −7.4954952531590871478835848659, −6.42369455375014219795894443161, −5.50942143540776093463578851282, −4.700076222736298029965660913849, −4.080082099179643414639766171121, −3.24348318942855038060493167902, −2.519019448086829746824140931662, −2.19554641443771411240140516135, −1.19753008543257363277651535374, 0.02928585740106334572866319935, 0.77999504868026461030023787873, 1.92819941551095166771451847338, 2.950358534637917197153225794993, 3.65769646046610917486435879552, 4.28033351445032549147306194455, 4.92402115170486669888929552726, 6.224884056897643162656819460319, 6.51852724954469526012766706226, 7.31402996587747676709022769989, 7.90657530044510007262238517317, 8.6151374014552229827399573392, 9.19523913409973351837212308875, 9.83621623622381738417831442802, 10.49898009660477574979291205427, 11.52390318580556998444314754649, 12.60707030389857009346605282365, 13.05370177187320808867418080655, 13.76605258587347283504205436266, 14.18618197553921862113074870999, 14.67570903094217246311709382382, 15.59471177189151889508268472626, 16.19703998357987686216505828356, 16.60738226731071673728685836460, 17.29224791571913483484453113708

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