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Properties

Label 1-4235-4235.72-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.977 - 0.212i$
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  + (−0.962 − 0.272i)2-s + (−0.207 + 0.978i)3-s + (0.851 + 0.524i)4-s + (0.466 − 0.884i)6-s + (−0.676 − 0.736i)8-s + (−0.913 − 0.406i)9-s + (−0.690 + 0.723i)12-s + (0.791 + 0.610i)13-s + (0.449 + 0.893i)16-s + (−0.424 + 0.905i)17-s + (0.768 + 0.640i)18-s + (0.123 + 0.992i)19-s + (−0.458 + 0.888i)23-s + (0.861 − 0.508i)24-s + (−0.595 − 0.803i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.962 − 0.272i)2-s + (−0.207 + 0.978i)3-s + (0.851 + 0.524i)4-s + (0.466 − 0.884i)6-s + (−0.676 − 0.736i)8-s + (−0.913 − 0.406i)9-s + (−0.690 + 0.723i)12-s + (0.791 + 0.610i)13-s + (0.449 + 0.893i)16-s + (−0.424 + 0.905i)17-s + (0.768 + 0.640i)18-s + (0.123 + 0.992i)19-s + (−0.458 + 0.888i)23-s + (0.861 − 0.508i)24-s + (−0.595 − 0.803i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.977 - 0.212i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.977 - 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06895642992 + 0.6404407745i\)
\(L(\frac12)\) \(\approx\) \(-0.06895642992 + 0.6404407745i\)
\(L(1)\) \(\approx\) \(0.5368257703 + 0.2907601092i\)
\(L(1)\) \(\approx\) \(0.5368257703 + 0.2907601092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.962 - 0.272i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (0.791 + 0.610i)T \)
17 \( 1 + (-0.424 + 0.905i)T \)
19 \( 1 + (0.123 + 0.992i)T \)
23 \( 1 + (-0.458 + 0.888i)T \)
29 \( 1 + (0.516 + 0.856i)T \)
31 \( 1 + (-0.179 + 0.983i)T \)
37 \( 1 + (-0.0760 + 0.997i)T \)
41 \( 1 + (-0.897 + 0.441i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (-0.768 + 0.640i)T \)
53 \( 1 + (-0.893 - 0.449i)T \)
59 \( 1 + (0.830 - 0.556i)T \)
61 \( 1 + (0.969 + 0.244i)T \)
67 \( 1 + (-0.371 + 0.928i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (-0.263 + 0.964i)T \)
79 \( 1 + (0.749 - 0.662i)T \)
83 \( 1 + (0.931 - 0.362i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (0.491 - 0.870i)T \)
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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

−17.94945115218851872275571056466, −17.620697832908131459329759910810, −16.77177001583162971916301553842, −16.161440858908493409693562281458, −15.4504428521575829500203308016, −14.77056175486574801101086763296, −13.770022315367855336300448100886, −13.40227619310909199072399320876, −12.41269683590807369436210006921, −11.72226179348579032031449837687, −11.14504775609094447038200360976, −10.53664374349687704279569799619, −9.61667148173092373185784335192, −8.85325021368078746638632651270, −8.22656709029250085800823790910, −7.6694879790341179586705450546, −6.792311231510814302086978654201, −6.43883597899814873252477557045, −5.57616937615613994970895725238, −4.88187604101436466270188628376, −3.4878935142191582033819995628, −2.48910425048336516385613403861, −2.04528966087982350455774110083, −0.85829324619708294365166760204, −0.3198711930408593770157643521, 1.289025359196873200124560175111, 1.886203330460291340902279960604, 3.24988911950524139715368514397, 3.517047411928398703603650085378, 4.46109482880516986575639757292, 5.424527497515898921418726333715, 6.32011267507471368106557069903, 6.7503360164002138636315887867, 8.0367201187691688991905806441, 8.46211351292446210982573233038, 9.10605883093861407808385978388, 10.01419233673353272655272572426, 10.23585490364302678348894680127, 11.19848610553240050050664210152, 11.541543487272915955126567875899, 12.3401115629856954138916299818, 13.18380329062770862759060688435, 14.19171025074547431609253855100, 14.86386112756870615464779015852, 15.637627165102131829169923659709, 16.20635314327909313338391413109, 16.597182979113441835676746548625, 17.478193860647389312850149225524, 17.887157391557606908120188172569, 18.72656199744984427061136117509

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