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Properties

Label 1-4235-4235.2553-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: $\chi_{4235} (2553, \cdot )$
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

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

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