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Properties

Label 1-2432-2432.611-r0-0-0
Degree $1$
Conductor $2432$
Sign $0.190 - 0.981i$
Analytic cond. $11.2941$
Root an. cond. $11.2941$
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  + (−0.876 + 0.480i)3-s + (−0.806 + 0.591i)5-s + (−0.130 + 0.991i)7-s + (0.537 − 0.843i)9-s + (0.751 − 0.659i)11-s + (−0.108 + 0.994i)13-s + (0.422 − 0.906i)15-s + (0.573 − 0.819i)17-s + (−0.362 − 0.932i)21-s + (0.0436 − 0.999i)23-s + (0.300 − 0.953i)25-s + (−0.0654 + 0.997i)27-s + (0.932 + 0.362i)29-s + (0.866 + 0.5i)31-s + (−0.342 + 0.939i)33-s + ⋯
L(s)  = 1  + (−0.876 + 0.480i)3-s + (−0.806 + 0.591i)5-s + (−0.130 + 0.991i)7-s + (0.537 − 0.843i)9-s + (0.751 − 0.659i)11-s + (−0.108 + 0.994i)13-s + (0.422 − 0.906i)15-s + (0.573 − 0.819i)17-s + (−0.362 − 0.932i)21-s + (0.0436 − 0.999i)23-s + (0.300 − 0.953i)25-s + (−0.0654 + 0.997i)27-s + (0.932 + 0.362i)29-s + (0.866 + 0.5i)31-s + (−0.342 + 0.939i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $0.190 - 0.981i$
Analytic conductor: \(11.2941\)
Root analytic conductor: \(11.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2432,\ (0:\ ),\ 0.190 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2783789747 - 0.2295803588i\)
\(L(\frac12)\) \(\approx\) \(0.2783789747 - 0.2295803588i\)
\(L(1)\) \(\approx\) \(0.6255120386 + 0.1495546811i\)
\(L(1)\) \(\approx\) \(0.6255120386 + 0.1495546811i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.876 + 0.480i)T \)
5 \( 1 + (-0.806 + 0.591i)T \)
7 \( 1 + (-0.130 + 0.991i)T \)
11 \( 1 + (0.751 - 0.659i)T \)
13 \( 1 + (-0.108 + 0.994i)T \)
17 \( 1 + (0.573 - 0.819i)T \)
23 \( 1 + (0.0436 - 0.999i)T \)
29 \( 1 + (0.932 + 0.362i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.831 - 0.555i)T \)
41 \( 1 + (-0.300 - 0.953i)T \)
43 \( 1 + (-0.988 - 0.152i)T \)
47 \( 1 + (-0.819 + 0.573i)T \)
53 \( 1 + (0.518 + 0.854i)T \)
59 \( 1 + (-0.915 - 0.402i)T \)
61 \( 1 + (-0.988 + 0.152i)T \)
67 \( 1 + (0.362 - 0.932i)T \)
71 \( 1 + (-0.999 + 0.0436i)T \)
73 \( 1 + (-0.953 + 0.300i)T \)
79 \( 1 + (0.0871 - 0.996i)T \)
83 \( 1 + (-0.0654 - 0.997i)T \)
89 \( 1 + (0.300 - 0.953i)T \)
97 \( 1 + (-0.984 + 0.173i)T \)
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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

−19.5586907424830630298770477751, −19.21925601517928756108240768221, −18.04022746507124066853458211393, −17.378158707484617394069478103801, −16.9487198468757540722494073209, −16.31863981384444713874749108036, −15.430154130748737041387845211527, −14.84420569390743146485796783831, −13.59940621195300750746491358442, −13.165795215699098968194444926376, −12.287467759234552473871456790523, −11.88258579602828470521868665700, −11.10220573462281329698710815795, −10.200671144732652492645195159565, −9.76221237549463438685594223558, −8.30076416482360273878721895840, −7.90252260459615169872997515347, −7.0740146620538532833948628981, −6.4541847733726916332006404605, −5.4382260918553632205256135795, −4.680315730781238983854853949341, −4.00832266252256900707717272292, −3.13136316527157097727379634189, −1.50981998198342249699657145420, −1.06631949768381079011607038190, 0.15964422647669395371914606527, 1.45237097848855504426461901380, 2.81380628565189216988224046221, 3.44885471865577241459887753193, 4.43306788606940658698239843526, 5.02947876070032622515719703807, 6.14187835841429114489270952786, 6.558529091384733436741006556381, 7.34152189812769303243604979778, 8.58436191617174749208489717982, 9.03928362508881931584801666292, 10.01408356305933807372361852105, 10.748421125171601630032492899605, 11.52582162042202062858481669628, 12.073275621587224993833708023844, 12.31460152957859666590659114227, 13.828887261126187355147755436446, 14.469200084147800929002442170737, 15.16061697554400498672771466355, 16.068921710503612683295138643695, 16.20563665694250476157492099804, 17.13881589988721546602836811297, 18.01825682615192095484448304658, 18.7854559458459132644285887765, 19.025202001810699520450274905363

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