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Properties

Label 1-4235-4235.1454-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.543 + 0.839i$
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  + (0.449 + 0.893i)2-s + (0.669 + 0.743i)3-s + (−0.595 + 0.803i)4-s + (−0.362 + 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.104 + 0.994i)9-s + (−0.995 + 0.0950i)12-s + (0.870 − 0.491i)13-s + (−0.290 − 0.956i)16-s + (0.179 − 0.983i)17-s + (−0.935 + 0.353i)18-s + (0.879 − 0.475i)19-s + (0.327 − 0.945i)23-s + (−0.532 − 0.846i)24-s + (0.830 + 0.556i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.449 + 0.893i)2-s + (0.669 + 0.743i)3-s + (−0.595 + 0.803i)4-s + (−0.362 + 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.104 + 0.994i)9-s + (−0.995 + 0.0950i)12-s + (0.870 − 0.491i)13-s + (−0.290 − 0.956i)16-s + (0.179 − 0.983i)17-s + (−0.935 + 0.353i)18-s + (0.879 − 0.475i)19-s + (0.327 − 0.945i)23-s + (−0.532 − 0.846i)24-s + (0.830 + 0.556i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.543 + 0.839i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.543 + 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.228791788 + 1.211439701i\)
\(L(\frac12)\) \(\approx\) \(2.228791788 + 1.211439701i\)
\(L(1)\) \(\approx\) \(1.284765199 + 0.8797804323i\)
\(L(1)\) \(\approx\) \(1.284765199 + 0.8797804323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.449 + 0.893i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.870 - 0.491i)T \)
17 \( 1 + (0.179 - 0.983i)T \)
19 \( 1 + (0.879 - 0.475i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.564 + 0.825i)T \)
31 \( 1 + (-0.749 - 0.662i)T \)
37 \( 1 + (-0.953 - 0.299i)T \)
41 \( 1 + (-0.254 - 0.967i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.935 - 0.353i)T \)
53 \( 1 + (0.290 - 0.956i)T \)
59 \( 1 + (0.710 + 0.703i)T \)
61 \( 1 + (0.548 + 0.836i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (-0.483 - 0.875i)T \)
79 \( 1 + (0.969 + 0.244i)T \)
83 \( 1 + (-0.0855 + 0.996i)T \)
89 \( 1 + (-0.235 - 0.971i)T \)
97 \( 1 + (-0.466 + 0.884i)T \)
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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

−18.49937342759350233435204793433, −17.856149835490260990136624971854, −17.21516447180398685130629893149, −16.01850954255723715253776096937, −15.382884465109416702942859940280, −14.46830045296380258269764159774, −14.18704889906921367926964262486, −13.265618568287963486153653227281, −13.04622747830700717301426408072, −12.11118411873784625631386270022, −11.61206358045592882228102365352, −10.90625115873994880840645261070, −9.92013045484394426172443435371, −9.44630163930900079148261094587, −8.54265425589960338134743109081, −8.101174360721456037315885059915, −7.03340918996136189067660762266, −6.260802664982465995905118441579, −5.64943605925694423117728018935, −4.65993223574543009972436049666, −3.57610295663752162378459972103, −3.436356198047106043343767174807, −2.37647219600028350239808560197, −1.482867273331347373119045360276, −1.15482029697104748565907944404, 0.57571509844455618604654495218, 2.11487913301101008068198685037, 3.12779898624815337581784702792, 3.48781595041456843789438022771, 4.38273401165494225720954756484, 5.21047817800907904538709378996, 5.52500821291298362527224311102, 6.755393564101722535446426612870, 7.27638050256364330921838027273, 8.14050266159259252791238507807, 8.75048652096977313531693929869, 9.24162613565914356859042346602, 10.10708221884498888136583533473, 10.87128688318807192528745332003, 11.723659841127312853603338679786, 12.56471869175938879133941062833, 13.42867731717000828659185712808, 13.80134467365632269842196784510, 14.47934090211141855202091345454, 15.172210168661876720636915219500, 15.71193770104098699330678005834, 16.343168464856449212167588841235, 16.709011150176211197980776064264, 17.8621256420582813022818319184, 18.23192722941507214784542435534

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