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Properties

Label 1-4235-4235.313-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.976 + 0.217i$
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  + (−0.132 + 0.991i)2-s + (−0.207 + 0.978i)3-s + (−0.964 − 0.263i)4-s + (−0.941 − 0.336i)6-s + (0.389 − 0.921i)8-s + (−0.913 − 0.406i)9-s + (0.458 − 0.888i)12-s + (0.931 + 0.362i)13-s + (0.861 + 0.508i)16-s + (−0.962 + 0.272i)17-s + (0.524 − 0.851i)18-s + (−0.830 − 0.556i)19-s + (−0.0950 − 0.995i)23-s + (0.820 + 0.572i)24-s + (−0.483 + 0.875i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.132 + 0.991i)2-s + (−0.207 + 0.978i)3-s + (−0.964 − 0.263i)4-s + (−0.941 − 0.336i)6-s + (0.389 − 0.921i)8-s + (−0.913 − 0.406i)9-s + (0.458 − 0.888i)12-s + (0.931 + 0.362i)13-s + (0.861 + 0.508i)16-s + (−0.962 + 0.272i)17-s + (0.524 − 0.851i)18-s + (−0.830 − 0.556i)19-s + (−0.0950 − 0.995i)23-s + (0.820 + 0.572i)24-s + (−0.483 + 0.875i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.976 + 0.217i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.976 + 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5623618928 + 0.06179189482i\)
\(L(\frac12)\) \(\approx\) \(0.5623618928 + 0.06179189482i\)
\(L(1)\) \(\approx\) \(0.5531238284 + 0.4557446896i\)
\(L(1)\) \(\approx\) \(0.5531238284 + 0.4557446896i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.132 + 0.991i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (0.931 + 0.362i)T \)
17 \( 1 + (-0.962 + 0.272i)T \)
19 \( 1 + (-0.830 - 0.556i)T \)
23 \( 1 + (-0.0950 - 0.995i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (-0.449 + 0.893i)T \)
37 \( 1 + (-0.353 + 0.935i)T \)
41 \( 1 + (0.564 + 0.825i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.524 - 0.851i)T \)
53 \( 1 + (0.508 + 0.861i)T \)
59 \( 1 + (-0.432 - 0.901i)T \)
61 \( 1 + (-0.380 + 0.924i)T \)
67 \( 1 + (-0.971 - 0.235i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (-0.299 - 0.953i)T \)
79 \( 1 + (0.290 + 0.956i)T \)
83 \( 1 + (-0.980 - 0.198i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
97 \( 1 + (0.996 + 0.0855i)T \)
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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

−18.352071215415836890086390734897, −17.8252090621225383614268418816, −17.40289202307384857788857082530, −16.542007173016248061336369894737, −15.71614984121946923893781471543, −14.68844084678762782916727958096, −14.00183734567570583857147664875, −13.222148393389147999455072488253, −13.04412116487320115473971976063, −12.135813361437924600330109856474, −11.55653158394290689220386244140, −10.919385311858114247089970106680, −10.38459089503259566569953847480, −9.30477121324992452712955919525, −8.71565802987177903467742725892, −8.03087554396810277468511045355, −7.394517628519398110736222439873, −6.32980188501273093121179711543, −5.77997167525020193499500836025, −4.8514936335209694502374736473, −3.96799031219006615204732414702, −3.18866425207695294092792721649, −2.29162223878666341903702638509, −1.70317198569811842106821853638, −0.83378058238193863173073948777, 0.21671718935144645861710478166, 1.495365065314863862217085961019, 2.81640723643267470810431595328, 3.729673491322705922865643160025, 4.51275350472502573393488496224, 4.82790515348007765800250313871, 5.90129980220188054480362893874, 6.44403417167770175933618852371, 6.996610143105295292316116128164, 8.358765486894333932491386608521, 8.618094898058354083062780846717, 9.1826884708253817632182215, 10.21756380758142348995118629029, 10.58494786708333133672816246270, 11.379117224093929967488709278314, 12.32314612053283192122473012950, 13.20670011194383174291118704028, 13.83103750218429086642829792047, 14.55233500016644058526924432711, 15.23165691981810918276850598813, 15.64879955130544748552337652194, 16.4230053882777446893212503581, 16.77568581454159986418852706165, 17.65546068226731909231788137793, 18.074867362794517093004244593545

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