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Properties

Label 1-4235-4235.3023-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.887 + 0.460i$
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  + (−0.0570 + 0.998i)2-s + (−0.951 − 0.309i)3-s + (−0.993 − 0.113i)4-s + (0.362 − 0.931i)6-s + (0.170 − 0.985i)8-s + (0.809 + 0.587i)9-s + (0.909 + 0.415i)12-s + (0.491 + 0.870i)13-s + (0.974 + 0.226i)16-s + (0.336 + 0.941i)17-s + (−0.633 + 0.774i)18-s + (−0.0285 + 0.999i)19-s + (0.755 − 0.654i)23-s + (−0.466 + 0.884i)24-s + (−0.897 + 0.441i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.0570 + 0.998i)2-s + (−0.951 − 0.309i)3-s + (−0.993 − 0.113i)4-s + (0.362 − 0.931i)6-s + (0.170 − 0.985i)8-s + (0.809 + 0.587i)9-s + (0.909 + 0.415i)12-s + (0.491 + 0.870i)13-s + (0.974 + 0.226i)16-s + (0.336 + 0.941i)17-s + (−0.633 + 0.774i)18-s + (−0.0285 + 0.999i)19-s + (0.755 − 0.654i)23-s + (−0.466 + 0.884i)24-s + (−0.897 + 0.441i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.887 + 0.460i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2592493327 + 1.063345385i\)
\(L(\frac12)\) \(\approx\) \(0.2592493327 + 1.063345385i\)
\(L(1)\) \(\approx\) \(0.6319857657 + 0.4509310895i\)
\(L(1)\) \(\approx\) \(0.6319857657 + 0.4509310895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0570 + 0.998i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (0.491 + 0.870i)T \)
17 \( 1 + (0.336 + 0.941i)T \)
19 \( 1 + (-0.0285 + 0.999i)T \)
23 \( 1 + (0.755 - 0.654i)T \)
29 \( 1 + (0.564 + 0.825i)T \)
31 \( 1 + (-0.198 + 0.980i)T \)
37 \( 1 + (0.676 + 0.736i)T \)
41 \( 1 + (0.254 + 0.967i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + (0.633 + 0.774i)T \)
53 \( 1 + (0.226 + 0.974i)T \)
59 \( 1 + (-0.254 + 0.967i)T \)
61 \( 1 + (0.998 - 0.0570i)T \)
67 \( 1 + (-0.540 - 0.841i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (-0.856 - 0.516i)T \)
79 \( 1 + (-0.696 + 0.717i)T \)
83 \( 1 + (-0.996 - 0.0855i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.884 - 0.466i)T \)
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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

−18.0406015916070076538700813249, −17.50958718692058600946357158190, −17.05578460422900053767879081823, −16.04033028509369109232765352625, −15.52822611269533311617679011004, −14.6841052306767452180946802407, −13.75314406105337061459380984086, −13.05714982597693549278455044849, −12.6318974853908109509774263776, −11.58432838079831516476251021918, −11.39917709481342488427221912026, −10.66430549528521976430543638356, −9.94342755099037387263306210035, −9.39835071675429200544839863011, −8.65310604954786744126593751654, −7.64270845345052002236639815390, −6.95496537459542219542064700322, −5.71042775171004223210237868857, −5.44480870446751439094744806558, −4.51629712800552169372583545068, −3.89208435613041076455480619157, −3.009473153049095093383701656136, −2.23275755064421612684059980553, −0.95151133364315205162634122811, −0.52870861893023798988265955292, 1.055448865767324942981127281511, 1.56487724996985687340210156861, 3.070660773811301731861210408917, 4.202524987328728170422438732770, 4.56600270043392599283888028455, 5.59498495589584530696989425589, 6.07027808183334070203173865684, 6.71203590157195832309563709503, 7.34472375030858051468069261589, 8.16861949970021316614547004772, 8.799927899993227694823214430518, 9.679553854451664397034076699806, 10.488428036420960868022973781752, 10.96808830213484897562660438140, 12.08770282126507514751770533435, 12.55424558722538794016852252286, 13.21009310940003519277618816196, 14.07386460519816136705431840525, 14.571506699887626162035797205117, 15.45638289418008026759118243596, 16.15382590894471849768084060132, 16.72030289482993638030642877110, 17.041009730664300271567005309112, 17.98138981529125910265452875643, 18.42079214675486213421368008625

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