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Properties

Label 1-4235-4235.1217-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.672 - 0.739i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.389 + 0.921i)2-s + (0.587 − 0.809i)3-s + (−0.696 − 0.717i)4-s + (0.516 + 0.856i)6-s + (0.931 − 0.362i)8-s + (−0.309 − 0.951i)9-s + (−0.989 + 0.142i)12-s + (−0.441 − 0.897i)13-s + (−0.0285 + 0.999i)16-s + (0.676 − 0.736i)17-s + (0.996 + 0.0855i)18-s + (−0.198 + 0.980i)19-s + (0.281 + 0.959i)23-s + (0.254 − 0.967i)24-s + (0.998 − 0.0570i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.389 + 0.921i)2-s + (0.587 − 0.809i)3-s + (−0.696 − 0.717i)4-s + (0.516 + 0.856i)6-s + (0.931 − 0.362i)8-s + (−0.309 − 0.951i)9-s + (−0.989 + 0.142i)12-s + (−0.441 − 0.897i)13-s + (−0.0285 + 0.999i)16-s + (0.676 − 0.736i)17-s + (0.996 + 0.0855i)18-s + (−0.198 + 0.980i)19-s + (0.281 + 0.959i)23-s + (0.254 − 0.967i)24-s + (0.998 − 0.0570i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.672 - 0.739i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.672 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896218675 - 0.8387158999i\)
\(L(\frac12)\) \(\approx\) \(1.896218675 - 0.8387158999i\)
\(L(1)\) \(\approx\) \(1.022981141 + 0.01251349031i\)
\(L(1)\) \(\approx\) \(1.022981141 + 0.01251349031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.389 + 0.921i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (-0.441 - 0.897i)T \)
17 \( 1 + (0.676 - 0.736i)T \)
19 \( 1 + (-0.198 + 0.980i)T \)
23 \( 1 + (0.281 + 0.959i)T \)
29 \( 1 + (-0.870 - 0.491i)T \)
31 \( 1 + (0.985 - 0.170i)T \)
37 \( 1 + (0.884 - 0.466i)T \)
41 \( 1 + (0.974 - 0.226i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.996 - 0.0855i)T \)
53 \( 1 + (-0.999 + 0.0285i)T \)
59 \( 1 + (0.974 + 0.226i)T \)
61 \( 1 + (-0.921 + 0.389i)T \)
67 \( 1 + (-0.755 - 0.654i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.791 - 0.610i)T \)
79 \( 1 + (0.774 + 0.633i)T \)
83 \( 1 + (0.825 + 0.564i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.967 + 0.254i)T \)
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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

−18.627724488565421999492904442396, −17.55938045821170000221121422242, −16.91788286696487588759434308673, −16.490197515102326645607068701273, −15.61591670608616613138308866408, −14.71492361381294779483008271168, −14.29621891087321103353515974199, −13.45808402595257001057779911583, −12.846393835239656379246233205893, −12.01722205695435456946609322252, −11.30105035480093881235296314419, −10.63400772073408706945783790452, −10.09814353475384169216215083675, −9.27995105134074531769212384156, −8.93136820747965419305951032833, −8.12203712446452622956627325753, −7.46862983009721001662505440785, −6.4628856817510258304432940556, −5.2536320697888998345485056886, −4.55375028878015570664220649056, −4.037061333163958325018426809732, −3.156471705804280898246158683335, −2.502995497596530707520644985443, −1.789651302491149326410712763319, −0.6873413431351497441620648699, 0.47029312793607399801017547839, 1.10692404617875571028928053738, 2.06985974237425346901456087673, 3.03536079193946159966858416831, 3.86012800949166446829126334001, 4.88384519403750954393804045403, 5.794754391237879721272099780036, 6.16068833655799112388383043933, 7.24351666859214690544354957223, 7.72845822545861656834072301143, 8.04753143743922697318777470905, 9.09205425140564217661436644327, 9.58663783608042049671645466972, 10.2603509715870265185444713976, 11.2864997639636294314320914505, 12.172569603569071735461576171153, 12.867212851734352367558895184693, 13.51262054153236237434536178299, 14.1361022696671533674609531832, 14.80510044069778024225252611675, 15.26110988906230405804453721773, 16.08486246362306578379817437511, 16.893675859752989121333002311155, 17.45952463554316607094474774997, 18.135078263063209737103900040561

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