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Properties

Label 1-6145-6145.1069-r1-0-0
Degree $1$
Conductor $6145$
Sign $0.988 - 0.148i$
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  + (0.576 + 0.817i)2-s + (0.453 − 0.891i)3-s + (−0.336 + 0.941i)4-s + (0.989 − 0.142i)6-s + (−0.383 − 0.923i)7-s + (−0.963 + 0.267i)8-s + (−0.588 − 0.808i)9-s + (−0.984 − 0.178i)11-s + (0.686 + 0.726i)12-s + (0.524 − 0.851i)13-s + (0.533 − 0.845i)14-s + (−0.774 − 0.633i)16-s + (0.777 + 0.629i)17-s + (0.321 − 0.946i)18-s + (0.888 − 0.458i)19-s + ⋯
L(s)  = 1  + (0.576 + 0.817i)2-s + (0.453 − 0.891i)3-s + (−0.336 + 0.941i)4-s + (0.989 − 0.142i)6-s + (−0.383 − 0.923i)7-s + (−0.963 + 0.267i)8-s + (−0.588 − 0.808i)9-s + (−0.984 − 0.178i)11-s + (0.686 + 0.726i)12-s + (0.524 − 0.851i)13-s + (0.533 − 0.845i)14-s + (−0.774 − 0.633i)16-s + (0.777 + 0.629i)17-s + (0.321 − 0.946i)18-s + (0.888 − 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $0.988 - 0.148i$
Analytic conductor: \(660.371\)
Root analytic conductor: \(660.371\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (1:\ ),\ 0.988 - 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.491803763 - 0.2611597743i\)
\(L(\frac12)\) \(\approx\) \(3.491803763 - 0.2611597743i\)
\(L(1)\) \(\approx\) \(1.513567179 + 0.06955451261i\)
\(L(1)\) \(\approx\) \(1.513567179 + 0.06955451261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (0.576 + 0.817i)T \)
3 \( 1 + (0.453 - 0.891i)T \)
7 \( 1 + (-0.383 - 0.923i)T \)
11 \( 1 + (-0.984 - 0.178i)T \)
13 \( 1 + (0.524 - 0.851i)T \)
17 \( 1 + (0.777 + 0.629i)T \)
19 \( 1 + (0.888 - 0.458i)T \)
23 \( 1 + (0.243 + 0.969i)T \)
29 \( 1 + (0.656 - 0.754i)T \)
31 \( 1 + (0.621 - 0.783i)T \)
37 \( 1 + (0.796 + 0.605i)T \)
41 \( 1 + (-0.355 + 0.934i)T \)
43 \( 1 + (0.925 - 0.379i)T \)
47 \( 1 + (0.326 + 0.945i)T \)
53 \( 1 + (-0.555 - 0.831i)T \)
59 \( 1 + (0.613 - 0.789i)T \)
61 \( 1 + (0.683 + 0.730i)T \)
67 \( 1 + (-0.840 + 0.542i)T \)
71 \( 1 + (0.898 + 0.439i)T \)
73 \( 1 + (-0.991 - 0.132i)T \)
79 \( 1 + (-0.267 + 0.963i)T \)
83 \( 1 + (0.906 + 0.421i)T \)
89 \( 1 + (0.117 - 0.993i)T \)
97 \( 1 + (-0.886 + 0.462i)T \)
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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

−17.80541710468464085297414559149, −16.42086516292852140973073482294, −16.11994113523624285936849715698, −15.5249446861259293479599216228, −14.82167550511438928478822409253, −14.13649892094551130727513002523, −13.7803829857308515952392837862, −12.86470534754589487200561938029, −12.206335707030308466719799090072, −11.68878389578920323648250465107, −10.81268268679665729496929888752, −10.31196415264347429787487604347, −9.669514748947941421561667552876, −9.03107444465907419555885895546, −8.559819616734229650773143898215, −7.573227362650748690241522029757, −6.50424373702148558312241557173, −5.66181979468042415681745401940, −5.17226659936405589271350034476, −4.54760019479546245149734805138, −3.69324458664375821810916693784, −2.95179641216642616942838962425, −2.60355608052292101477576847774, −1.711227009093129630615784458436, −0.579773151491958021480667373683, 0.585247751016624334541545456565, 1.125802287026586210200008124775, 2.555595646758603591539715188996, 3.10874382961019703311599753595, 3.666097232167050139154535631004, 4.57721798728135896323682423688, 5.586971379889353396413492229700, 5.97913769898413155169364004562, 6.7926058568866673812248797921, 7.49508075462650483341150042982, 7.98191902351414040306549250031, 8.316071731552313511143085869392, 9.483980998186787997434431569677, 10.01481113396443512868235430132, 11.10161053326623687783530626484, 11.745166658361918757895886954096, 12.63912767954307818216850136181, 13.20200196672967329875490393726, 13.44379978390974938360848078117, 14.08458866768402754145094084546, 14.83549336214401063706656636207, 15.54127966707702749992963755786, 16.01458875543023748088095974965, 16.867609445370880363616410753921, 17.59511411345000350571334723771

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