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Properties

Label 1-4235-4235.1277-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.0225 - 0.999i$
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  + (−0.998 + 0.0475i)2-s + (0.866 − 0.5i)3-s + (0.995 − 0.0950i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 − 0.866i)9-s + (0.814 − 0.580i)12-s + (0.909 + 0.415i)13-s + (0.981 − 0.189i)16-s + (0.971 − 0.235i)17-s + (−0.458 + 0.888i)18-s + (0.235 − 0.971i)19-s + (−0.189 − 0.981i)23-s + (−0.786 + 0.618i)24-s + (−0.928 − 0.371i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.998 + 0.0475i)2-s + (0.866 − 0.5i)3-s + (0.995 − 0.0950i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 − 0.866i)9-s + (0.814 − 0.580i)12-s + (0.909 + 0.415i)13-s + (0.981 − 0.189i)16-s + (0.971 − 0.235i)17-s + (−0.458 + 0.888i)18-s + (0.235 − 0.971i)19-s + (−0.189 − 0.981i)23-s + (−0.786 + 0.618i)24-s + (−0.928 − 0.371i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0225 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0225 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.0225 - 0.999i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.0225 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.205877134 - 1.178950281i\)
\(L(\frac12)\) \(\approx\) \(1.205877134 - 1.178950281i\)
\(L(1)\) \(\approx\) \(0.9908185986 - 0.3105382949i\)
\(L(1)\) \(\approx\) \(0.9908185986 - 0.3105382949i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.998 + 0.0475i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (0.971 - 0.235i)T \)
19 \( 1 + (0.235 - 0.971i)T \)
23 \( 1 + (-0.189 - 0.981i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (-0.580 + 0.814i)T \)
37 \( 1 + (0.0950 - 0.995i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (0.458 + 0.888i)T \)
53 \( 1 + (0.189 - 0.981i)T \)
59 \( 1 + (0.0475 - 0.998i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
67 \( 1 + (0.458 - 0.888i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.945 - 0.327i)T \)
79 \( 1 + (0.786 + 0.618i)T \)
83 \( 1 + (-0.755 - 0.654i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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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

−18.65253621375316369501055170190, −18.043000520376295731859375249710, −17.091834453003172633523594739849, −16.48009097093751241603683085811, −15.94719228828042393455976459528, −15.20181127763745687437590101635, −14.772091275099489540716264074324, −13.81984003641304891304492012864, −13.23191899543577756695789963404, −12.20584504519585871286128111941, −11.618696226790175471243441520613, −10.645582390516144378829311139403, −10.14544800966439910606803506472, −9.68650776898396797716541256044, −8.72261879865481708178464708956, −8.332712837826018203073248096740, −7.67364644289683623018849503237, −6.99645911510507307774986394271, −5.903537834334058778420823193364, −5.356933849035816740001600489903, −3.993960730894680306730583375490, −3.43584759685173199830231055824, −2.77223004221187505088218009913, −1.71460083105277836380450021397, −1.16086600852176572449694139038, 0.61303410289686683348658418573, 1.400336763893941488994739535210, 2.17660329759210300025676538574, 3.015778309472315791439494104394, 3.58991019486051483273124046503, 4.76024570523996039024695254933, 5.87564955353197247103611452799, 6.68040644898021762910344288092, 7.07340892358601901439520133173, 8.03971857311238896430301447592, 8.44348848439623992557040081457, 9.12986711602340807974082050690, 9.757032358097603297185450425187, 10.492671221589448863345354845470, 11.31416353890922730803193580242, 12.016917857984655653331805052261, 12.6901891697006672432592223917, 13.49908202738277963200682507668, 14.32584500553957682561771879602, 14.73220529325706659688872662126, 15.783287737274636304425663781304, 16.0512917642173284130404129332, 16.924999255092120724839582282813, 17.85425797157557197265230304249, 18.18688774229340286597466283061

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