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Properties

Label 1-4275-4275.2767-r0-0-0
Degree $1$
Conductor $4275$
Sign $0.990 + 0.139i$
Analytic cond. $19.8530$
Root an. cond. $19.8530$
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  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.866 − 0.5i)7-s + (−0.951 + 0.309i)8-s + (−0.104 − 0.994i)11-s + (0.406 + 0.913i)13-s + (−0.809 + 0.587i)14-s + (−0.104 + 0.994i)16-s + (0.207 − 0.978i)17-s + (−0.951 − 0.309i)22-s + (−0.994 + 0.104i)23-s + 26-s + (0.207 + 0.978i)28-s + (0.309 − 0.951i)29-s + (−0.669 + 0.743i)31-s + (0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.866 − 0.5i)7-s + (−0.951 + 0.309i)8-s + (−0.104 − 0.994i)11-s + (0.406 + 0.913i)13-s + (−0.809 + 0.587i)14-s + (−0.104 + 0.994i)16-s + (0.207 − 0.978i)17-s + (−0.951 − 0.309i)22-s + (−0.994 + 0.104i)23-s + 26-s + (0.207 + 0.978i)28-s + (0.309 − 0.951i)29-s + (−0.669 + 0.743i)31-s + (0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4275\)    =    \(3^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.990 + 0.139i$
Analytic conductor: \(19.8530\)
Root analytic conductor: \(19.8530\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4275} (2767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4275,\ (0:\ ),\ 0.990 + 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6798378180 + 0.04748633129i\)
\(L(\frac12)\) \(\approx\) \(0.6798378180 + 0.04748633129i\)
\(L(1)\) \(\approx\) \(0.7664389551 - 0.4886733906i\)
\(L(1)\) \(\approx\) \(0.7664389551 - 0.4886733906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.406 - 0.913i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (0.406 + 0.913i)T \)
17 \( 1 + (0.207 - 0.978i)T \)
23 \( 1 + (-0.994 + 0.104i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.207 - 0.978i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.207 + 0.978i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.994 + 0.104i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.207 + 0.978i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−18.33187120140135607429390954378, −17.48167417543578449117109541940, −17.00038350980609262528360725361, −16.058392982657607851276437302674, −15.6767043062986223254295333241, −15.05377923042260539491901890384, −14.45312977991101885483487828863, −13.596787068690805047141465721223, −12.830397134397922560435187101474, −12.51120231888589785013517179725, −11.902260522278731055532663302675, −10.56422305174115056204907309792, −10.09694002920283156492787321769, −9.10402777923682895680152214586, −8.70112290914097126408010886403, −7.67001619956336014188934680541, −7.294782718602737610734979319539, −6.2105922365945433575392458414, −5.899954198720205397881896303527, −5.12477546475885786285788978352, −4.15961966573740915536151104012, −3.55474682337905916040555500314, −2.752767298743831527673205754758, −1.75216260809111882336433503584, −0.198222587070332568582989418724, 0.84135906990674938828040729874, 1.708683220961517901574125337743, 2.78578924233774905414133870153, 3.30327110826156368629775764055, 4.06381770595999826257695057393, 4.73287043812059017642013371704, 5.77527968513417625562113854630, 6.26001623088008949316770908773, 7.08988873908974163309908460700, 8.14877823403038350159396496399, 8.94476166470022045669665702603, 9.58279553209299590936091189732, 10.17674548603879073041438958227, 10.9279142562293173782194326312, 11.59893842806548316602042653219, 12.12081473532558667275743773448, 13.00788917659625499566164470828, 13.73421690598281753134062355285, 13.877182289006531372552400762174, 14.738264754087310361194569637666, 15.888163223649696700888803694064, 16.15264698888529393812137224314, 17.00017038605132940357426467744, 17.94292319099934847672778141329, 18.598783053918293428947234553997

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