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Properties

Label 1-4235-4235.734-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.920 - 0.391i$
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  + (−0.985 − 0.170i)2-s + (−0.809 + 0.587i)3-s + (0.941 + 0.336i)4-s + (0.897 − 0.441i)6-s + (−0.870 − 0.491i)8-s + (0.309 − 0.951i)9-s + (−0.959 + 0.281i)12-s + (0.0285 − 0.999i)13-s + (0.774 + 0.633i)16-s + (−0.516 + 0.856i)17-s + (−0.466 + 0.884i)18-s + (0.0855 − 0.996i)19-s + (−0.841 + 0.540i)23-s + (0.993 − 0.113i)24-s + (−0.198 + 0.980i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.985 − 0.170i)2-s + (−0.809 + 0.587i)3-s + (0.941 + 0.336i)4-s + (0.897 − 0.441i)6-s + (−0.870 − 0.491i)8-s + (0.309 − 0.951i)9-s + (−0.959 + 0.281i)12-s + (0.0285 − 0.999i)13-s + (0.774 + 0.633i)16-s + (−0.516 + 0.856i)17-s + (−0.466 + 0.884i)18-s + (0.0855 − 0.996i)19-s + (−0.841 + 0.540i)23-s + (0.993 − 0.113i)24-s + (−0.198 + 0.980i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.920 - 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5326702249 - 0.1086530592i\)
\(L(\frac12)\) \(\approx\) \(0.5326702249 - 0.1086530592i\)
\(L(1)\) \(\approx\) \(0.5008605725 + 0.01814711065i\)
\(L(1)\) \(\approx\) \(0.5008605725 + 0.01814711065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.985 - 0.170i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.0285 - 0.999i)T \)
17 \( 1 + (-0.516 + 0.856i)T \)
19 \( 1 + (0.0855 - 0.996i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
31 \( 1 + (0.564 - 0.825i)T \)
37 \( 1 + (-0.610 - 0.791i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.466 - 0.884i)T \)
53 \( 1 + (-0.774 + 0.633i)T \)
59 \( 1 + (-0.696 + 0.717i)T \)
61 \( 1 + (-0.985 + 0.170i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (0.998 + 0.0570i)T \)
79 \( 1 + (0.736 + 0.676i)T \)
83 \( 1 + (0.254 - 0.967i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.993 - 0.113i)T \)
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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

−18.33007615116376453507215726063, −17.82617206579596650577088584636, −17.08576963116021814558753282958, −16.54102733731070819165612421176, −16.00305079894873060924469258524, −15.37254690755233516244096664574, −14.1280789769419165745108134491, −13.93923194963811468738525146660, −12.645349981386549169856139033759, −12.12023679551615707883311895998, −11.50575328600683038114664600935, −10.89400999342995225938231081821, −10.19215018398258511559370360077, −9.468199034186617111650051049937, −8.71597293076690611821842463344, −7.873405165886371869068425314923, −7.33845111959917530043321438344, −6.516945817920555687585515961065, −6.15494087759379388466868222591, −5.22248440572981522746904208739, −4.42940725593594763163588351677, −3.22843167276823177699755388529, −2.04736320461770067386952429181, −1.727985535245112284141444082103, −0.57996577412739302719842900599, 0.40816254896080275360203172955, 1.36008727845733521741908869639, 2.367820534321726130391787071902, 3.311563203317998280847543886, 4.03004211957775485908734372657, 4.98679910171374756455709890136, 5.96836572256071156710791858953, 6.26755041767902124728195183193, 7.35194734654339771502782630692, 7.89504768628025220835328226125, 8.884714482761053502862847646671, 9.41178888646920750015782068625, 10.15975016318833165576498807132, 10.74612334734812570592803854035, 11.25722976910997779745763980880, 11.93196499209891076496830395328, 12.722061270338395583549073006461, 13.28010485476996194093318792428, 14.62586312528850187988470062458, 15.34943419341375639874373607427, 15.64942424623342238053997681821, 16.46412231912568280470542301669, 17.079809607524322206889489810134, 17.79011588222017434010831256812, 17.94505142266319489263538127526

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