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Properties

Label 1-4235-4235.1297-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.456 + 0.889i$
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  + (0.690 − 0.723i)2-s + (0.866 + 0.5i)3-s + (−0.0475 − 0.998i)4-s + (0.959 − 0.281i)6-s + (−0.755 − 0.654i)8-s + (0.5 + 0.866i)9-s + (0.458 − 0.888i)12-s + (0.540 + 0.841i)13-s + (−0.995 + 0.0950i)16-s + (−0.618 + 0.786i)17-s + (0.971 + 0.235i)18-s + (−0.786 + 0.618i)19-s + (0.0950 + 0.995i)23-s + (−0.327 − 0.945i)24-s + (0.981 + 0.189i)26-s + i·27-s + ⋯
L(s)  = 1  + (0.690 − 0.723i)2-s + (0.866 + 0.5i)3-s + (−0.0475 − 0.998i)4-s + (0.959 − 0.281i)6-s + (−0.755 − 0.654i)8-s + (0.5 + 0.866i)9-s + (0.458 − 0.888i)12-s + (0.540 + 0.841i)13-s + (−0.995 + 0.0950i)16-s + (−0.618 + 0.786i)17-s + (0.971 + 0.235i)18-s + (−0.786 + 0.618i)19-s + (0.0950 + 0.995i)23-s + (−0.327 − 0.945i)24-s + (0.981 + 0.189i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.456 + 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.112221229 + 1.291058480i\)
\(L(\frac12)\) \(\approx\) \(2.112221229 + 1.291058480i\)
\(L(1)\) \(\approx\) \(1.726153499 - 0.1279134036i\)
\(L(1)\) \(\approx\) \(1.726153499 - 0.1279134036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.690 - 0.723i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.540 + 0.841i)T \)
17 \( 1 + (-0.618 + 0.786i)T \)
19 \( 1 + (-0.786 + 0.618i)T \)
23 \( 1 + (0.0950 + 0.995i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.888 + 0.458i)T \)
37 \( 1 + (0.998 + 0.0475i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.971 + 0.235i)T \)
53 \( 1 + (-0.0950 + 0.995i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (0.971 + 0.235i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.814 + 0.580i)T \)
79 \( 1 + (-0.327 + 0.945i)T \)
83 \( 1 + (-0.909 + 0.415i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (-0.755 - 0.654i)T \)
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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

−18.04147283534037321545170066749, −17.80618868839020345922588431118, −16.693168679536622504563564848938, −16.093919513489869621525256633119, −15.38276238431464459823713759480, −14.72691762306408016541213347854, −14.37464754031909247577525977699, −13.39729401710902119686101950035, −12.97882822746818119829311472528, −12.61889096950529274191808484432, −11.523497675775487256398513873368, −10.9260707926935916176964055087, −9.7489404683461999826271885105, −8.94304382756797249759612545386, −8.44263329002685817283324982033, −7.78271128854123042914181301923, −7.01304209066060355688810356100, −6.50074812330605373013244639774, −5.7156280533637589547900349256, −4.72383880507705329548099549075, −4.130005720044133093883016772434, −3.14009300824615460216752017737, −2.724140924041676396898152039913, −1.767478935026882325004906428, −0.4130016821073536290351505094, 1.42722064550612116124484116330, 1.945198312605759178578553578895, 2.74348005704176150055114523740, 3.72461909378372917970171481093, 4.04213909013524794578663730412, 4.76426358364101383461827360199, 5.75680856547283887051752065773, 6.3978681489297318456292647582, 7.37269869428538303378883290870, 8.316062379454467894276937603656, 9.00284471816432868825817065852, 9.59623261008207631510045984352, 10.29766870217840205627189152733, 11.04846009900973796731751257069, 11.47732079899022262408265889465, 12.63872715436254641299601280643, 13.0109005132334509702310083089, 13.886260806670127313294384975755, 14.19349611705975735678760128146, 15.16241465951828663022759754096, 15.37022881035915357524229875832, 16.29084106699236162887190078047, 17.0077433709992234880610042851, 18.10237158315667877884942595116, 18.7947472786497188546852804711

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