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Properties

Label 1-4235-4235.2809-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.943 + 0.332i$
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  + (0.830 + 0.556i)2-s + (−0.913 + 0.406i)3-s + (0.380 + 0.924i)4-s + (−0.985 − 0.170i)6-s + (−0.198 + 0.980i)8-s + (0.669 − 0.743i)9-s + (−0.723 − 0.690i)12-s + (0.564 + 0.825i)13-s + (−0.710 + 0.703i)16-s + (−0.123 + 0.992i)17-s + (0.969 − 0.244i)18-s + (−0.683 − 0.730i)19-s + (0.888 + 0.458i)23-s + (−0.217 − 0.976i)24-s + (0.00951 + 0.999i)26-s + (−0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.830 + 0.556i)2-s + (−0.913 + 0.406i)3-s + (0.380 + 0.924i)4-s + (−0.985 − 0.170i)6-s + (−0.198 + 0.980i)8-s + (0.669 − 0.743i)9-s + (−0.723 − 0.690i)12-s + (0.564 + 0.825i)13-s + (−0.710 + 0.703i)16-s + (−0.123 + 0.992i)17-s + (0.969 − 0.244i)18-s + (−0.683 − 0.730i)19-s + (0.888 + 0.458i)23-s + (−0.217 − 0.976i)24-s + (0.00951 + 0.999i)26-s + (−0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.943 + 0.332i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.943 + 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3374012665 + 1.971111203i\)
\(L(\frac12)\) \(\approx\) \(0.3374012665 + 1.971111203i\)
\(L(1)\) \(\approx\) \(1.016435363 + 0.8254548285i\)
\(L(1)\) \(\approx\) \(1.016435363 + 0.8254548285i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.830 + 0.556i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.564 + 0.825i)T \)
17 \( 1 + (-0.123 + 0.992i)T \)
19 \( 1 + (-0.683 - 0.730i)T \)
23 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (0.0855 - 0.996i)T \)
31 \( 1 + (0.879 + 0.475i)T \)
37 \( 1 + (0.761 + 0.647i)T \)
41 \( 1 + (-0.466 + 0.884i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.969 + 0.244i)T \)
53 \( 1 + (0.710 + 0.703i)T \)
59 \( 1 + (0.999 + 0.0380i)T \)
61 \( 1 + (-0.0665 - 0.997i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (0.516 + 0.856i)T \)
73 \( 1 + (0.625 + 0.780i)T \)
79 \( 1 + (-0.398 - 0.917i)T \)
83 \( 1 + (-0.774 + 0.633i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + (0.736 - 0.676i)T \)
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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

−18.30819900424908205684932116494, −17.48165037147543301481616693183, −16.57124417664388437182070302056, −16.081014928358574532374384238041, −15.2930482660952038397121913032, −14.630108683914116248546985588012, −13.75342265603763993569914551288, −13.131296885239182853095327340928, −12.67625747190606162745648852883, −11.9255267025877761528073767335, −11.38855693660232617473265092073, −10.57967635326513265189442959101, −10.31207375795862242877782343547, −9.27995270924608491139397528874, −8.30846730787995274013084424811, −7.308421601257412157917468873827, −6.71248090334237050999317922056, −5.94721080842538418939255200369, −5.39112742987844028909790573099, −4.690962735870575841377940519806, −3.94311634016195743713851523923, −2.969703558370182582519882628317, −2.2185871326682094254686016592, −1.229667844596362601965313323091, −0.531276250360354374654617932292, 1.110793059606891125483849797653, 2.19693893516826876031049157454, 3.22693112618815886391292738688, 4.121193632975700131438799565373, 4.48376453054528966958775976083, 5.31901683713614605371986754054, 6.09564693058178316570353834527, 6.57806168754872530774306349127, 7.17735248435166727819205124083, 8.27374123599393461538705816866, 8.86527770434153256714490681130, 9.80842558511347165244907964754, 10.71620923015742025205341047678, 11.311350394384351139817887569340, 11.82699253790520153387154420535, 12.61192796759127927169389865032, 13.26451391941718932498906979483, 13.83265540209992132336482753600, 14.883381992006360586191760910995, 15.3328298356927334239034448277, 15.819386026343542820619194563, 16.77120080732937355637157879765, 17.049513739608930281068961199119, 17.63640995214524956276298093801, 18.54626683211861402453873412539

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