Properties

Label 1-4235-4235.1433-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.917 + 0.398i$
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  + (−0.475 − 0.879i)2-s + (−0.994 + 0.104i)3-s + (−0.548 + 0.836i)4-s + (0.564 + 0.825i)6-s + (0.996 + 0.0855i)8-s + (0.978 − 0.207i)9-s + (0.458 − 0.888i)12-s + (−0.967 + 0.254i)13-s + (−0.398 − 0.917i)16-s + (0.938 + 0.345i)17-s + (−0.647 − 0.761i)18-s + (0.272 − 0.962i)19-s + (−0.0950 − 0.995i)23-s + (−0.999 + 0.0190i)24-s + (0.683 + 0.730i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.475 − 0.879i)2-s + (−0.994 + 0.104i)3-s + (−0.548 + 0.836i)4-s + (0.564 + 0.825i)6-s + (0.996 + 0.0855i)8-s + (0.978 − 0.207i)9-s + (0.458 − 0.888i)12-s + (−0.967 + 0.254i)13-s + (−0.398 − 0.917i)16-s + (0.938 + 0.345i)17-s + (−0.647 − 0.761i)18-s + (0.272 − 0.962i)19-s + (−0.0950 − 0.995i)23-s + (−0.999 + 0.0190i)24-s + (0.683 + 0.730i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.917 + 0.398i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.917 + 0.398i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06234876687 - 0.3002911071i\)
\(L(\frac12)\) \(\approx\) \(-0.06234876687 - 0.3002911071i\)
\(L(1)\) \(\approx\) \(0.4835817224 - 0.2310459501i\)
\(L(1)\) \(\approx\) \(0.4835817224 - 0.2310459501i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.475 - 0.879i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (-0.967 + 0.254i)T \)
17 \( 1 + (0.938 + 0.345i)T \)
19 \( 1 + (0.272 - 0.962i)T \)
23 \( 1 + (-0.0950 - 0.995i)T \)
29 \( 1 + (0.466 - 0.884i)T \)
31 \( 1 + (-0.161 - 0.986i)T \)
37 \( 1 + (0.780 + 0.625i)T \)
41 \( 1 + (-0.610 + 0.791i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (0.647 - 0.761i)T \)
53 \( 1 + (-0.917 - 0.398i)T \)
59 \( 1 + (-0.991 + 0.132i)T \)
61 \( 1 + (0.851 - 0.524i)T \)
67 \( 1 + (-0.971 - 0.235i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (-0.999 - 0.00951i)T \)
79 \( 1 + (-0.797 - 0.603i)T \)
83 \( 1 + (0.676 + 0.736i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
97 \( 1 + (-0.856 + 0.516i)T \)
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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

−18.51399311264313883130119256434, −17.93702232826101081147965877183, −17.34222974095649181272821759773, −16.77123958846215314138899232575, −16.1375545183259147578733296938, −15.69264188203811308795903758134, −14.70971277994899606675929855824, −14.252542462515151910140446815601, −13.39025500302915819329606485333, −12.49011245110834925588995968160, −12.04777151923682164670354634988, −11.08728432383711631830486069162, −10.34759491500637099422685488054, −9.852848485288577864461784895491, −9.20314487147119009991564417052, −8.110434036784792245723737521335, −7.48002241583240819884336036558, −7.03090876445388655188024839100, −6.10764425044380108198349027942, −5.47707412193221861751351396795, −5.02591908733291762534010721994, −4.163388294155011351410154802476, −3.09612206623276441221186723384, −1.68041212780864207349835402779, −1.08754285394073519870391307074, 0.15280144902509764339608953249, 0.98056686425224204256614361533, 1.97297523351936910789375380963, 2.76919136423547996288673285150, 3.72883774803522426319793843772, 4.60414258348761393591116882246, 4.988534000042651270082923859945, 6.06107306027997114817898036640, 6.864807740437007966925234775974, 7.62483348571344414123848102622, 8.30642655924911276458274575186, 9.37865415467715278101864840659, 9.85036998190885240518918435633, 10.42817307273417734028841470492, 11.17634955659878534112582095132, 11.886665360021285211382289695289, 12.17600264712384808826167748429, 13.04036083720898669175465643134, 13.59691922382082299474237023073, 14.64869117411253148298530661636, 15.35723427851591883220819627095, 16.36798088909494464815423642090, 16.82558685164934167601513428051, 17.318584831713078850080194926539, 17.996700122887703537864454649368

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