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Properties

Label 1-4235-4235.1018-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.998 - 0.0511i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.924 + 0.380i)2-s + (−0.743 − 0.669i)3-s + (0.710 − 0.703i)4-s + (0.941 + 0.336i)6-s + (−0.389 + 0.921i)8-s + (0.104 + 0.994i)9-s + (−0.998 + 0.0475i)12-s + (−0.931 − 0.362i)13-s + (0.00951 − 0.999i)16-s + (−0.244 + 0.969i)17-s + (−0.475 − 0.879i)18-s + (0.0665 − 0.997i)19-s + (−0.814 + 0.580i)23-s + (0.905 − 0.424i)24-s + (0.999 − 0.0190i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.924 + 0.380i)2-s + (−0.743 − 0.669i)3-s + (0.710 − 0.703i)4-s + (0.941 + 0.336i)6-s + (−0.389 + 0.921i)8-s + (0.104 + 0.994i)9-s + (−0.998 + 0.0475i)12-s + (−0.931 − 0.362i)13-s + (0.00951 − 0.999i)16-s + (−0.244 + 0.969i)17-s + (−0.475 − 0.879i)18-s + (0.0665 − 0.997i)19-s + (−0.814 + 0.580i)23-s + (0.905 − 0.424i)24-s + (0.999 − 0.0190i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.998 - 0.0511i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1018, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.998 - 0.0511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2852559039 + 0.007306653338i\)
\(L(\frac12)\) \(\approx\) \(0.2852559039 + 0.007306653338i\)
\(L(1)\) \(\approx\) \(0.4310889233 + 0.02989908541i\)
\(L(1)\) \(\approx\) \(0.4310889233 + 0.02989908541i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.924 + 0.380i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (-0.931 - 0.362i)T \)
17 \( 1 + (-0.244 + 0.969i)T \)
19 \( 1 + (0.0665 - 0.997i)T \)
23 \( 1 + (-0.814 + 0.580i)T \)
29 \( 1 + (-0.985 - 0.170i)T \)
31 \( 1 + (-0.548 - 0.836i)T \)
37 \( 1 + (0.986 - 0.161i)T \)
41 \( 1 + (-0.564 - 0.825i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (-0.475 + 0.879i)T \)
53 \( 1 + (-0.999 + 0.00951i)T \)
59 \( 1 + (0.997 + 0.0760i)T \)
61 \( 1 + (-0.991 + 0.132i)T \)
67 \( 1 + (0.690 - 0.723i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (-0.976 - 0.217i)T \)
79 \( 1 + (-0.683 + 0.730i)T \)
83 \( 1 + (0.980 + 0.198i)T \)
89 \( 1 + (-0.786 - 0.618i)T \)
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

−18.23882992104163474358769236264, −17.53212024675635086667869304858, −16.829686265238097168513705594764, −16.331054230232776342175588384161, −15.91987532629691748363199518181, −14.90587971429404544200459682872, −14.424542001523213072319937229988, −13.21858059736461341125807173075, −12.3434437828780869704488467561, −11.933888186778905396824365467868, −11.291377580204373817870153746502, −10.57717131679969552736800614404, −9.91226205506706449337894367561, −9.47362548155248260793199067349, −8.74992359437860405046175105245, −7.81856209031775435677320591574, −7.103055220656165629523036331131, −6.41887980580370864017480073033, −5.58076009344937273476095910151, −4.70921500119518906145145626260, −3.9338982467500082190470265275, −3.14069967083481124937628412707, −2.21119017841507966826275420009, −1.3179014047356716965484822226, −0.21512779675099315081447165407, 0.26099458616263611326208418264, 1.29733805946749450673579562151, 2.05376456679613259576327149064, 2.742687833253557659946314341457, 4.15390471056286673645359998949, 5.10543699523748079340702758644, 5.79496639382523643989091937817, 6.32508849341080366876293549196, 7.219276428762581037719899603507, 7.65769148275205394846485936171, 8.2879810826929898320803362130, 9.318100352351672262516970205566, 9.848488630405444594940873811894, 10.755358180561920553047848669798, 11.205965874721328314273003191433, 11.889638185376866660455451305692, 12.71293471300893890799651471982, 13.29347568287284289210936885085, 14.289773444385897848220659821710, 14.96697620749993696863034365267, 15.67718399675643013612715637014, 16.35186778147347836766385711577, 17.163924885576917904023297637997, 17.41772404105633626367960790039, 18.051756388073241142955888243816

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