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Properties

Label 1-6145-6145.1033-r1-0-0
Degree $1$
Conductor $6145$
Sign $-0.305 + 0.952i$
Analytic cond. $660.371$
Root an. cond. $660.371$
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  + (−0.963 + 0.267i)2-s + (−0.987 + 0.157i)3-s + (0.856 − 0.516i)4-s + (0.909 − 0.416i)6-s + (0.379 − 0.925i)7-s + (−0.686 + 0.726i)8-s + (0.950 − 0.311i)9-s + (−0.511 + 0.859i)11-s + (−0.764 + 0.644i)12-s + (−0.996 − 0.0868i)13-s + (−0.117 + 0.993i)14-s + (0.467 − 0.884i)16-s + (0.453 − 0.891i)17-s + (−0.831 + 0.555i)18-s + (−0.989 − 0.142i)19-s + ⋯
L(s)  = 1  + (−0.963 + 0.267i)2-s + (−0.987 + 0.157i)3-s + (0.856 − 0.516i)4-s + (0.909 − 0.416i)6-s + (0.379 − 0.925i)7-s + (−0.686 + 0.726i)8-s + (0.950 − 0.311i)9-s + (−0.511 + 0.859i)11-s + (−0.764 + 0.644i)12-s + (−0.996 − 0.0868i)13-s + (−0.117 + 0.993i)14-s + (0.467 − 0.884i)16-s + (0.453 − 0.891i)17-s + (−0.831 + 0.555i)18-s + (−0.989 − 0.142i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $-0.305 + 0.952i$
Analytic conductor: \(660.371\)
Root analytic conductor: \(660.371\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (1:\ ),\ -0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05177466333 + 0.07098044870i\)
\(L(\frac12)\) \(\approx\) \(0.05177466333 + 0.07098044870i\)
\(L(1)\) \(\approx\) \(0.4463896999 + 0.01340922684i\)
\(L(1)\) \(\approx\) \(0.4463896999 + 0.01340922684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (-0.963 + 0.267i)T \)
3 \( 1 + (-0.987 + 0.157i)T \)
7 \( 1 + (0.379 - 0.925i)T \)
11 \( 1 + (-0.511 + 0.859i)T \)
13 \( 1 + (-0.996 - 0.0868i)T \)
17 \( 1 + (0.453 - 0.891i)T \)
19 \( 1 + (-0.989 - 0.142i)T \)
23 \( 1 + (0.671 + 0.740i)T \)
29 \( 1 + (0.546 - 0.837i)T \)
31 \( 1 + (-0.904 - 0.425i)T \)
37 \( 1 + (-0.369 + 0.929i)T \)
41 \( 1 + (-0.886 + 0.462i)T \)
43 \( 1 + (0.919 + 0.393i)T \)
47 \( 1 + (0.542 - 0.840i)T \)
53 \( 1 + (-0.193 - 0.981i)T \)
59 \( 1 + (-0.917 - 0.397i)T \)
61 \( 1 + (-0.774 + 0.633i)T \)
67 \( 1 + (-0.989 + 0.147i)T \)
71 \( 1 + (0.979 - 0.203i)T \)
73 \( 1 + (0.388 - 0.921i)T \)
79 \( 1 + (-0.726 + 0.686i)T \)
83 \( 1 + (0.262 + 0.964i)T \)
89 \( 1 + (0.345 - 0.938i)T \)
97 \( 1 + (0.991 + 0.127i)T \)
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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

−17.19453974960815696214180367154, −16.97246319585756796364816887359, −16.18976179657464131641781952730, −15.59898550239324825622342259228, −14.92963551886749348790414808098, −14.169487142631149333467943314048, −12.85283982900675870746961555593, −12.47569783093216274503519409677, −12.112827676019781839275975198726, −11.17569325179030798186321291441, −10.60294827830390668386373860072, −10.406655609782625152664101964358, −9.17568257818706461847010857731, −8.80184081165521580867439652156, −7.94470227922887832354801577554, −7.37239007809924083831724097730, −6.51988489419773880908675396481, −5.91142095815934033092605754280, −5.28582153988062167233460143028, −4.4395896610651968879669897284, −3.366471958025168920465835411195, −2.48717602937492452835565507690, −1.84420813184389082488323572336, −1.01008715951030289873307370305, −0.0383658207695245862308513334, 0.46970609245915368866890847824, 1.40856747363761933685530339354, 2.128763318490299160729877529136, 3.13862902528646066251171605350, 4.349949118022089938064039039723, 4.92463996268197731588220480224, 5.49358451889645664023789609423, 6.494034940427936310148505660063, 7.105659318487510557955701698443, 7.50213750801446690807121923779, 8.17419476965118541327028021694, 9.37960883848632204257388751649, 9.78203445671765789199698409052, 10.411751115908688623129939995124, 10.90093309410758796367265324034, 11.66124422477347937104744842318, 12.169075827578269255772233162401, 12.99973222007290763998816108260, 13.81409265819547957830733649407, 14.85854959726767701771304739880, 15.167935940604870699169705350272, 15.91191085537776275029463163970, 16.768439491548212257240147449989, 17.03168381107283045553622243839, 17.56138840181052875491417536316

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