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Properties

Label 1-4235-4235.3019-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.720 + 0.693i$
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  + (−0.988 + 0.151i)2-s + (0.978 + 0.207i)3-s + (0.953 − 0.299i)4-s + (−0.998 − 0.0570i)6-s + (−0.897 + 0.441i)8-s + (0.913 + 0.406i)9-s + (0.995 − 0.0950i)12-s + (−0.198 − 0.980i)13-s + (0.820 − 0.572i)16-s + (0.991 − 0.132i)17-s + (−0.964 − 0.263i)18-s + (−0.432 + 0.901i)19-s + (0.327 − 0.945i)23-s + (−0.969 + 0.244i)24-s + (0.345 + 0.938i)26-s + (0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.988 + 0.151i)2-s + (0.978 + 0.207i)3-s + (0.953 − 0.299i)4-s + (−0.998 − 0.0570i)6-s + (−0.897 + 0.441i)8-s + (0.913 + 0.406i)9-s + (0.995 − 0.0950i)12-s + (−0.198 − 0.980i)13-s + (0.820 − 0.572i)16-s + (0.991 − 0.132i)17-s + (−0.964 − 0.263i)18-s + (−0.432 + 0.901i)19-s + (0.327 − 0.945i)23-s + (−0.969 + 0.244i)24-s + (0.345 + 0.938i)26-s + (0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.720 + 0.693i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.720 + 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.637083272 + 0.6605137557i\)
\(L(\frac12)\) \(\approx\) \(1.637083272 + 0.6605137557i\)
\(L(1)\) \(\approx\) \(1.049757767 + 0.1739831380i\)
\(L(1)\) \(\approx\) \(1.049757767 + 0.1739831380i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.988 + 0.151i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.198 - 0.980i)T \)
17 \( 1 + (0.991 - 0.132i)T \)
19 \( 1 + (-0.432 + 0.901i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (-0.0285 + 0.999i)T \)
31 \( 1 + (0.861 - 0.508i)T \)
37 \( 1 + (0.595 + 0.803i)T \)
41 \( 1 + (0.774 + 0.633i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (-0.964 + 0.263i)T \)
53 \( 1 + (-0.820 - 0.572i)T \)
59 \( 1 + (0.161 + 0.986i)T \)
61 \( 1 + (-0.625 + 0.780i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (0.941 + 0.336i)T \)
73 \( 1 + (-0.123 + 0.992i)T \)
79 \( 1 + (-0.532 + 0.846i)T \)
83 \( 1 + (-0.974 + 0.226i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (-0.696 - 0.717i)T \)
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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

−18.59002475793753719592529334865, −17.4888857787640885951921403157, −17.25812919684198162393371156868, −16.182314638413796004468283450698, −15.705330489032839312394596180224, −14.97898016213763346832136061933, −14.306879033772113604506941960237, −13.58054339271746301343111498651, −12.76562332737242900014277178601, −12.083940912995491463522822648716, −11.40638074527661758220598717147, −10.584996352879861084317365212119, −9.73980193276124723505608031243, −9.335367469618550596042194440560, −8.689772979953712870402207012111, −7.91916948024367123504254529599, −7.37668365921479050243142657879, −6.71408187921518132591694469733, −5.941517047054231122973380229630, −4.69981254115134330599555842540, −3.788468295028012953216421922838, −3.07050201912433612934891490524, −2.26553263785721153994245212522, −1.65138814942495222358602439818, −0.69064934193106426039799636447, 0.95314952681642515863917528330, 1.63835641302921133076091857859, 2.89346282808091201483328509560, 2.92159881460808130169167748543, 4.16698341761402825107080438308, 5.109516620212747714168451935223, 6.01822362170654399917009233706, 6.78530520327364526011484421341, 7.70995271744676378214229175780, 8.08598056212750156850953420032, 8.6283036060506872128000102840, 9.63362714251301631128344530725, 9.95646678657845357348872609197, 10.62052661393808841260805473038, 11.39649363063328701653810264737, 12.50697168041319670314047346083, 12.79569352051002889391158454176, 13.97767011299833154100121149244, 14.69966344449139133585435141205, 14.96653294669335729635801725303, 15.8338568659139565560354169164, 16.49609391359909829272631204230, 16.95661920615762574080237266739, 18.03075963692063087068668276992, 18.444787715764682835651037909753

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