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Properties

Label 1-4235-4235.114-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.969 + 0.245i$
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  + (−0.123 − 0.992i)2-s + (−0.913 + 0.406i)3-s + (−0.969 + 0.244i)4-s + (0.516 + 0.856i)6-s + (0.362 + 0.931i)8-s + (0.669 − 0.743i)9-s + (0.786 − 0.618i)12-s + (−0.897 + 0.441i)13-s + (0.879 − 0.475i)16-s + (−0.953 + 0.299i)17-s + (−0.820 − 0.572i)18-s + (−0.948 + 0.318i)19-s + (−0.723 − 0.690i)23-s + (−0.710 − 0.703i)24-s + (0.548 + 0.836i)26-s + (−0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.123 − 0.992i)2-s + (−0.913 + 0.406i)3-s + (−0.969 + 0.244i)4-s + (0.516 + 0.856i)6-s + (0.362 + 0.931i)8-s + (0.669 − 0.743i)9-s + (0.786 − 0.618i)12-s + (−0.897 + 0.441i)13-s + (0.879 − 0.475i)16-s + (−0.953 + 0.299i)17-s + (−0.820 − 0.572i)18-s + (−0.948 + 0.318i)19-s + (−0.723 − 0.690i)23-s + (−0.710 − 0.703i)24-s + (0.548 + 0.836i)26-s + (−0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.969 + 0.245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3729626327 + 0.04643219551i\)
\(L(\frac12)\) \(\approx\) \(0.3729626327 + 0.04643219551i\)
\(L(1)\) \(\approx\) \(0.5008276518 - 0.1759738462i\)
\(L(1)\) \(\approx\) \(0.5008276518 - 0.1759738462i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.123 - 0.992i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.897 + 0.441i)T \)
17 \( 1 + (-0.953 + 0.299i)T \)
19 \( 1 + (-0.948 + 0.318i)T \)
23 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (-0.870 - 0.491i)T \)
31 \( 1 + (0.345 - 0.938i)T \)
37 \( 1 + (0.532 - 0.846i)T \)
41 \( 1 + (0.974 - 0.226i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.820 + 0.572i)T \)
53 \( 1 + (-0.879 - 0.475i)T \)
59 \( 1 + (-0.683 + 0.730i)T \)
61 \( 1 + (0.797 + 0.603i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.380 - 0.924i)T \)
79 \( 1 + (0.161 - 0.986i)T \)
83 \( 1 + (0.564 - 0.825i)T \)
89 \( 1 + (-0.995 + 0.0950i)T \)
97 \( 1 + (0.254 - 0.967i)T \)
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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

−18.117182881781663617506562313, −17.55101133433987764267629365370, −17.06332211187667242941066389795, −16.43664197700442851777175318616, −15.66779861073168649076738255449, −15.20004970566710953531524771174, −14.33010649657138693800007628340, −13.60764006845164307628966117030, −12.85026126740969754717727033888, −12.51100570921360455631906932497, −11.43665368212362243831907253109, −10.84717251036042655651776182619, −9.936420605663867486419289760715, −9.454763652716757312277381102626, −8.36424096969297796776415211146, −7.83946469616059301978486938517, −6.997590752610444386162149018669, −6.550808906901352507533654833460, −5.838769666554866869913501914465, −4.939230509637258904808486222361, −4.68015742755272145294022806055, −3.597296565058152681259976003, −2.327038820769947968885045711363, −1.377531471094221398489013049545, −0.23225391801409840602920015339, 0.534555166838328398529862766970, 1.86872149557440805958086124989, 2.32262910084148882343377137188, 3.55129156810430103831851550696, 4.42035560599281167021557341734, 4.52975478019742645748544372338, 5.70173181895245903422871010136, 6.286605337478496706429956857113, 7.27963003588897042145594946270, 8.14941444888140345460447046782, 9.03705062420739399742341231164, 9.64656816617495951564711121867, 10.25941100229950758758651484379, 10.931549400164687341920945529945, 11.48565533029618722738183676206, 12.12523374791736506773838893943, 12.79972059805729475375987244424, 13.28451841139103427945089599910, 14.434711417457904642266037035, 14.8612761431086393941516601792, 15.81264686083412392896011395052, 16.69717118471004359575861760070, 17.08616513824245427041156387633, 17.77128367487548235294857445041, 18.35483404375201589716884590044

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