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Properties

Label 1-4235-4235.1357-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.988 + 0.150i$
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  + (0.441 + 0.897i)2-s + (0.587 − 0.809i)3-s + (−0.610 + 0.791i)4-s + (0.985 + 0.170i)6-s + (−0.980 − 0.198i)8-s + (−0.309 − 0.951i)9-s + (0.281 + 0.959i)12-s + (0.825 − 0.564i)13-s + (−0.254 − 0.967i)16-s + (0.389 + 0.921i)17-s + (0.717 − 0.696i)18-s + (0.974 − 0.226i)19-s + (−0.540 − 0.841i)23-s + (−0.736 + 0.676i)24-s + (0.870 + 0.491i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.441 + 0.897i)2-s + (0.587 − 0.809i)3-s + (−0.610 + 0.791i)4-s + (0.985 + 0.170i)6-s + (−0.980 − 0.198i)8-s + (−0.309 − 0.951i)9-s + (0.281 + 0.959i)12-s + (0.825 − 0.564i)13-s + (−0.254 − 0.967i)16-s + (0.389 + 0.921i)17-s + (0.717 − 0.696i)18-s + (0.974 − 0.226i)19-s + (−0.540 − 0.841i)23-s + (−0.736 + 0.676i)24-s + (0.870 + 0.491i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.622595374 + 0.1985662687i\)
\(L(\frac12)\) \(\approx\) \(2.622595374 + 0.1985662687i\)
\(L(1)\) \(\approx\) \(1.524277793 + 0.2801485579i\)
\(L(1)\) \(\approx\) \(1.524277793 + 0.2801485579i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.441 + 0.897i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (0.825 - 0.564i)T \)
17 \( 1 + (0.389 + 0.921i)T \)
19 \( 1 + (0.974 - 0.226i)T \)
23 \( 1 + (-0.540 - 0.841i)T \)
29 \( 1 + (-0.0855 + 0.996i)T \)
31 \( 1 + (0.0285 + 0.999i)T \)
37 \( 1 + (0.336 + 0.941i)T \)
41 \( 1 + (0.466 - 0.884i)T \)
43 \( 1 + (0.909 - 0.415i)T \)
47 \( 1 + (-0.717 - 0.696i)T \)
53 \( 1 + (0.967 + 0.254i)T \)
59 \( 1 + (-0.466 - 0.884i)T \)
61 \( 1 + (-0.897 - 0.441i)T \)
67 \( 1 + (0.989 - 0.142i)T \)
71 \( 1 + (0.516 + 0.856i)T \)
73 \( 1 + (-0.931 - 0.362i)T \)
79 \( 1 + (-0.993 - 0.113i)T \)
83 \( 1 + (0.633 + 0.774i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.676 + 0.736i)T \)
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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

−18.46647205229459581894663160570, −17.98205918530346118585999371659, −16.91298743861715927188093837745, −16.04076292623257952137600037516, −15.691051279804795392793397154144, −14.74201667248891342235194712514, −14.21260454125850485122420921147, −13.592145712782286942063821836792, −13.16462352055057749582071993262, −12.00045086984222667586071827898, −11.45415592079685802593218838840, −10.95811745094175364102232480684, −9.98031920032979385072397323608, −9.52850919222741493723876003644, −9.0759773710162126490534082076, −8.07545236729241516760867505984, −7.41976927860796581278980169193, −6.00811247807758265998308177923, −5.61657598860502770025392242948, −4.59828859478524391168356984341, −4.09324489533286483797429713616, −3.36535259721853990127981265689, −2.693056705011101433740380277859, −1.88426730840731522659270278611, −0.90478530602193218235194797828, 0.713538035327350060979404196678, 1.67363450002910057999079066685, 2.84734988115198193361153096457, 3.40806231125078842431197218993, 4.11489591827021584876915427733, 5.22429397659260319443348789023, 5.86954617910083116834186318768, 6.56581513151314338544479711193, 7.19457070089883879004876536813, 7.96466002087702728044396269857, 8.4676453996024917583041099974, 9.032909464237912223302163821327, 9.97692193974922262446675225263, 10.916021201362462677612091556133, 11.967639400858792552730272515157, 12.49119492841455580768546197140, 13.050070140586360351861904933846, 13.77959976529756820544815859804, 14.26886827779268008035611826943, 14.889687743198682267065604161288, 15.65284855575526248245377639894, 16.18400411922134219201798605900, 17.11288555506483182767839936116, 17.68494442794814099817376746362, 18.418536863261265090165673346209

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