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Properties

Label 1-4235-4235.2047-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.00338 + 0.999i$
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  + (0.0950 − 0.995i)2-s + (0.866 − 0.5i)3-s + (−0.981 − 0.189i)4-s + (−0.415 − 0.909i)6-s + (−0.281 + 0.959i)8-s + (0.5 − 0.866i)9-s + (−0.945 + 0.327i)12-s + (−0.755 − 0.654i)13-s + (0.928 + 0.371i)16-s + (−0.458 + 0.888i)17-s + (−0.814 − 0.580i)18-s + (−0.888 + 0.458i)19-s + (0.371 − 0.928i)23-s + (0.235 + 0.971i)24-s + (−0.723 + 0.690i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.0950 − 0.995i)2-s + (0.866 − 0.5i)3-s + (−0.981 − 0.189i)4-s + (−0.415 − 0.909i)6-s + (−0.281 + 0.959i)8-s + (0.5 − 0.866i)9-s + (−0.945 + 0.327i)12-s + (−0.755 − 0.654i)13-s + (0.928 + 0.371i)16-s + (−0.458 + 0.888i)17-s + (−0.814 − 0.580i)18-s + (−0.888 + 0.458i)19-s + (0.371 − 0.928i)23-s + (0.235 + 0.971i)24-s + (−0.723 + 0.690i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00338 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00338 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.00338 + 0.999i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.00338 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3447023452 - 0.3435382777i\)
\(L(\frac12)\) \(\approx\) \(-0.3447023452 - 0.3435382777i\)
\(L(1)\) \(\approx\) \(0.7468415954 - 0.7174800627i\)
\(L(1)\) \(\approx\) \(0.7468415954 - 0.7174800627i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0950 - 0.995i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (-0.458 + 0.888i)T \)
19 \( 1 + (-0.888 + 0.458i)T \)
23 \( 1 + (0.371 - 0.928i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (0.327 - 0.945i)T \)
37 \( 1 + (0.189 + 0.981i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (-0.281 + 0.959i)T \)
47 \( 1 + (0.814 - 0.580i)T \)
53 \( 1 + (-0.371 - 0.928i)T \)
59 \( 1 + (-0.995 + 0.0950i)T \)
61 \( 1 + (-0.580 - 0.814i)T \)
67 \( 1 + (0.814 + 0.580i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.618 - 0.786i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.989 - 0.142i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
97 \( 1 + (-0.281 + 0.959i)T \)
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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

−18.78812922974052207646339249313, −18.23564733406730461224607022372, −17.20487176106239608180711544976, −16.85906071049476920271208504896, −15.94312659358051101533353974562, −15.52983917068573138532677251973, −14.866531009230215783316436492926, −14.22880996744114150861363658796, −13.73574233598775303553566396124, −13.03526005432406010315530490630, −12.32971431680497530312274908554, −11.27872570991550218085504630051, −10.44274775424291629566749471233, −9.585970965909487435341480454032, −9.07568006038678815125528920676, −8.6650280139162886847104402179, −7.54696738370002462923338691999, −7.282750006714805536475804776136, −6.44977326110917482935275095765, −5.38206245030286447672094072217, −4.74920467875639864574110295674, −4.202174963365686241693781768932, −3.31159969275349755279716478026, −2.540075032229524004262185136917, −1.49687440651699108610200703955, 0.1108598547965199709991164665, 1.2124830282300004631576874757, 2.14882632692687402602469918435, 2.520895049326154198119359773799, 3.48552584525336958146639081496, 4.10855506099881957952221088924, 4.867315813764364546248708995003, 5.931574216508122529839778637758, 6.64051712465245105562926155796, 7.75157193990347236336006967927, 8.25386679018258350560079408964, 8.8677799726260876786806740155, 9.71871158789840308021260462683, 10.22135551972107510642982433020, 11.00101729437347955924943518717, 11.84673724797735327838139756168, 12.683377286023366661316272955952, 12.86258480491960369071061832648, 13.62815924295777852535023056845, 14.40000338747271937523772052442, 15.011364337250853560302660096390, 15.380637098796471999862567547189, 16.96632790538048932003628056450, 17.20109166364654812909023200418, 18.17387085560407065306889923792

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