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Properties

Label 1-4335-4335.2948-r1-0-0
Degree $1$
Conductor $4335$
Sign $-0.943 + 0.331i$
Analytic cond. $465.860$
Root an. cond. $465.860$
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  + (−0.228 + 0.973i)2-s + (−0.895 − 0.445i)4-s + (−0.0230 + 0.999i)7-s + (0.638 − 0.769i)8-s + (0.656 + 0.754i)11-s + (0.0922 + 0.995i)13-s + (−0.967 − 0.251i)14-s + (0.602 + 0.798i)16-s + (−0.228 − 0.973i)19-s + (−0.884 + 0.466i)22-s + (0.999 + 0.0230i)23-s + (−0.990 − 0.138i)26-s + (0.466 − 0.884i)28-s + (0.0692 − 0.997i)29-s + (−0.884 + 0.466i)31-s + (−0.914 + 0.403i)32-s + ⋯
L(s)  = 1  + (−0.228 + 0.973i)2-s + (−0.895 − 0.445i)4-s + (−0.0230 + 0.999i)7-s + (0.638 − 0.769i)8-s + (0.656 + 0.754i)11-s + (0.0922 + 0.995i)13-s + (−0.967 − 0.251i)14-s + (0.602 + 0.798i)16-s + (−0.228 − 0.973i)19-s + (−0.884 + 0.466i)22-s + (0.999 + 0.0230i)23-s + (−0.990 − 0.138i)26-s + (0.466 − 0.884i)28-s + (0.0692 − 0.997i)29-s + (−0.884 + 0.466i)31-s + (−0.914 + 0.403i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4335 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4335 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4335\)    =    \(3 \cdot 5 \cdot 17^{2}\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(465.860\)
Root analytic conductor: \(465.860\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4335} (2948, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4335,\ (1:\ ),\ -0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3299404514 + 1.936479987i\)
\(L(\frac12)\) \(\approx\) \(0.3299404514 + 1.936479987i\)
\(L(1)\) \(\approx\) \(0.7494593478 + 0.6099270925i\)
\(L(1)\) \(\approx\) \(0.7494593478 + 0.6099270925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.228 + 0.973i)T \)
7 \( 1 + (-0.0230 + 0.999i)T \)
11 \( 1 + (0.656 + 0.754i)T \)
13 \( 1 + (0.0922 + 0.995i)T \)
19 \( 1 + (-0.228 - 0.973i)T \)
23 \( 1 + (0.999 + 0.0230i)T \)
29 \( 1 + (0.0692 - 0.997i)T \)
31 \( 1 + (-0.884 + 0.466i)T \)
37 \( 1 + (0.620 + 0.783i)T \)
41 \( 1 + (-0.978 - 0.206i)T \)
43 \( 1 + (0.990 - 0.138i)T \)
47 \( 1 + (0.932 + 0.361i)T \)
53 \( 1 + (-0.403 + 0.914i)T \)
59 \( 1 + (0.998 - 0.0461i)T \)
61 \( 1 + (0.940 - 0.339i)T \)
67 \( 1 + (0.526 - 0.850i)T \)
71 \( 1 + (0.690 + 0.723i)T \)
73 \( 1 + (-0.506 + 0.862i)T \)
79 \( 1 + (0.811 - 0.584i)T \)
83 \( 1 + (-0.824 + 0.565i)T \)
89 \( 1 + (0.995 + 0.0922i)T \)
97 \( 1 + (-0.690 - 0.723i)T \)
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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

−17.881359657164077964659564435005, −17.29964885347846163598016665557, −16.62843907343657680197647006827, −16.18559109082542697107787723041, −14.82707913639417899014534355197, −14.41055717264907898759491458790, −13.61431349009865786037808996654, −12.99150307532869993013354582082, −12.49826768716715977672762115555, −11.54178672429703398190604283892, −10.91994167582142903687797766991, −10.492711911383606215750331263554, −9.74467280982643053498644242368, −8.97702332328280174321627247353, −8.29911678340289770140107153974, −7.60636040274287508317556967995, −6.83627923423066976060797947452, −5.75441604283670390126501159385, −5.08580202596572424181596403799, −4.00588167730826764455772192407, −3.631514909435590969631177332948, −2.898355273185502799524771375679, −1.81593769025300314619660314758, −0.96200931233032357229798313765, −0.46150239759593961652623997157, 0.782654521921775525231332349777, 1.75979267553846247959222793319, 2.59379469426251517845399728216, 3.80263680707383773769270970679, 4.54484406949377499633455112270, 5.15912902376431757125399053418, 5.98964502925710776645984546247, 6.73784628322102356447440850991, 7.124265454198714063524613520020, 8.12621340415848157763784467561, 8.9235988230189319668015795281, 9.24783196362413883465537053573, 9.88221702942491090811901674723, 10.969753636576673404433250276692, 11.657155807501943862699338120343, 12.47649462818752313587918120395, 13.10132111110851525471195512409, 13.94438760657892477959678149572, 14.56501490428657000660448660369, 15.238359858642314745536378349075, 15.615067227196322518043200602537, 16.44101430752782189684599317071, 17.216488386133851944802306763931, 17.52741889055295507069745406850, 18.49957039030226389399887500924

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