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Properties

Label 1-4235-4235.258-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.203 - 0.979i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.226 − 0.974i)2-s + (0.951 − 0.309i)3-s + (−0.897 − 0.441i)4-s + (−0.0855 − 0.996i)6-s + (−0.633 + 0.774i)8-s + (0.809 − 0.587i)9-s + (−0.989 − 0.142i)12-s + (−0.884 + 0.466i)13-s + (0.610 + 0.791i)16-s + (0.980 + 0.198i)17-s + (−0.389 − 0.921i)18-s + (0.993 + 0.113i)19-s + (0.281 − 0.959i)23-s + (−0.362 + 0.931i)24-s + (0.254 + 0.967i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.226 − 0.974i)2-s + (0.951 − 0.309i)3-s + (−0.897 − 0.441i)4-s + (−0.0855 − 0.996i)6-s + (−0.633 + 0.774i)8-s + (0.809 − 0.587i)9-s + (−0.989 − 0.142i)12-s + (−0.884 + 0.466i)13-s + (0.610 + 0.791i)16-s + (0.980 + 0.198i)17-s + (−0.389 − 0.921i)18-s + (0.993 + 0.113i)19-s + (0.281 − 0.959i)23-s + (−0.362 + 0.931i)24-s + (0.254 + 0.967i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.203 - 0.979i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.203 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.094239704 - 1.704355012i\)
\(L(\frac12)\) \(\approx\) \(2.094239704 - 1.704355012i\)
\(L(1)\) \(\approx\) \(1.317584283 - 0.8306193632i\)
\(L(1)\) \(\approx\) \(1.317584283 - 0.8306193632i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.226 - 0.974i)T \)
3 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (-0.884 + 0.466i)T \)
17 \( 1 + (0.980 + 0.198i)T \)
19 \( 1 + (0.993 + 0.113i)T \)
23 \( 1 + (0.281 - 0.959i)T \)
29 \( 1 + (0.736 + 0.676i)T \)
31 \( 1 + (-0.696 - 0.717i)T \)
37 \( 1 + (-0.170 + 0.985i)T \)
41 \( 1 + (-0.516 + 0.856i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (0.389 - 0.921i)T \)
53 \( 1 + (0.791 + 0.610i)T \)
59 \( 1 + (0.516 + 0.856i)T \)
61 \( 1 + (-0.974 + 0.226i)T \)
67 \( 1 + (-0.755 + 0.654i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (0.825 + 0.564i)T \)
79 \( 1 + (0.998 - 0.0570i)T \)
83 \( 1 + (0.336 - 0.941i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.931 - 0.362i)T \)
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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

−18.3986644917376811773682447421, −17.773875859172906903901282122691, −17.050753071271061507739146844849, −16.28606919554378103733751830894, −15.70861449269828741869513185432, −15.170713433256647697977772162048, −14.46914284222784833972687310141, −13.92941966268026869282302336323, −13.455807284077803133434128604496, −12.45275775513282059876941557715, −12.101776705907413475605236768337, −10.76774612111089198847039761286, −9.92573045221303433884994310418, −9.440346032922189816329024844449, −8.80872237251284825204414148340, −7.89944950135396584264526901739, −7.47969712224552759857971395358, −6.95056599924605821334826591084, −5.6786461343380332514958249043, −5.21689431356467851191609347208, −4.45410456641398997068662951734, −3.46572169473468280546065112161, −3.13442155048522919365358343519, −2.02269356369326351749960373228, −0.76851300083548402499119446537, 0.910941919434417572822783025413, 1.565619883196902221832893244, 2.54910849428275715151100318922, 2.988571679576771926158788385473, 3.815857889234621576383329727421, 4.57271542038931104758260930092, 5.30670507884355753087465814815, 6.31102876035225990191183111215, 7.241950469625489616024737293670, 7.92553087022206672887686545557, 8.7207112495619005314641986052, 9.330847862906332603015095286060, 10.013158590631753102936033918582, 10.481601624466577134252474301818, 11.67371122735873930462226530225, 12.10613209143930652976445949380, 12.73355197214920663698115175434, 13.48781023286125675893030758020, 14.03654051100519433176786058958, 14.761540643715439890394448281654, 15.0076198014300803888539690537, 16.25788501766149503056393032893, 16.9126302288243970447529871237, 17.94474885803014078363229930357, 18.45579564684764618791845426932

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