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Properties

Label 1-4235-4235.194-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.204 - 0.978i$
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  + (−0.999 − 0.0190i)2-s + (0.913 − 0.406i)3-s + (0.999 + 0.0380i)4-s + (−0.921 + 0.389i)6-s + (−0.998 − 0.0570i)8-s + (0.669 − 0.743i)9-s + (0.928 − 0.371i)12-s + (0.985 − 0.170i)13-s + (0.997 + 0.0760i)16-s + (0.398 − 0.917i)17-s + (−0.683 + 0.730i)18-s + (0.861 + 0.508i)19-s + (−0.235 − 0.971i)23-s + (−0.935 + 0.353i)24-s + (−0.988 + 0.151i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.999 − 0.0190i)2-s + (0.913 − 0.406i)3-s + (0.999 + 0.0380i)4-s + (−0.921 + 0.389i)6-s + (−0.998 − 0.0570i)8-s + (0.669 − 0.743i)9-s + (0.928 − 0.371i)12-s + (0.985 − 0.170i)13-s + (0.997 + 0.0760i)16-s + (0.398 − 0.917i)17-s + (−0.683 + 0.730i)18-s + (0.861 + 0.508i)19-s + (−0.235 − 0.971i)23-s + (−0.935 + 0.353i)24-s + (−0.988 + 0.151i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.204 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043997060 - 1.285193612i\)
\(L(\frac12)\) \(\approx\) \(1.043997060 - 1.285193612i\)
\(L(1)\) \(\approx\) \(0.9821891643 - 0.3420352968i\)
\(L(1)\) \(\approx\) \(0.9821891643 - 0.3420352968i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 - 0.0190i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.985 - 0.170i)T \)
17 \( 1 + (0.398 - 0.917i)T \)
19 \( 1 + (0.861 + 0.508i)T \)
23 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (-0.198 - 0.980i)T \)
31 \( 1 + (0.0665 - 0.997i)T \)
37 \( 1 + (-0.272 + 0.962i)T \)
41 \( 1 + (0.0855 + 0.996i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.683 - 0.730i)T \)
53 \( 1 + (-0.997 + 0.0760i)T \)
59 \( 1 + (-0.820 - 0.572i)T \)
61 \( 1 + (0.483 - 0.875i)T \)
67 \( 1 + (-0.981 + 0.189i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (0.179 - 0.983i)T \)
79 \( 1 + (-0.964 + 0.263i)T \)
83 \( 1 + (0.0285 - 0.999i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (0.774 + 0.633i)T \)
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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

−18.58337273201043308130628647790, −17.953763632483086623909918020443, −17.331055948690416100968643426832, −16.3444945649430403803854005741, −15.897481867578528346802846753127, −15.47605781543790063930876404363, −14.4965348117069080915025523935, −14.094946768857889910867036571620, −13.10583715706669984038636853455, −12.44193381286761851993062213826, −11.44854263300273897066543782900, −10.80028847352083502245056301670, −10.25547825320308047292163977139, −9.416402270082600484630162301874, −8.936815959561612932222298411856, −8.35448127972334979599441037057, −7.55372551979194097569446452315, −7.081550715476704993086761860326, −6.02136951877741962108787237188, −5.32905023181533304666073931803, −4.129204245114053561401440044889, −3.37048441295083575139968385020, −2.82932281721866224099105993315, −1.65252646790846207046791727140, −1.28798147003340496141085366056, 0.570064402131369106456290906129, 1.3798285558746141585579460735, 2.16933485467266779084083439156, 3.02301997514917178195430154407, 3.53791715366289990885387053335, 4.64643476677252540456464080968, 5.93629153965129405911744424404, 6.40175842818610479933551410556, 7.34589741106783307936855234358, 7.888217310858335268080416793272, 8.39994286845913348108697272764, 9.20695232628405437192923789095, 9.752774774239401795326762624198, 10.366574559780465351616818625343, 11.38771073158196761221103754272, 11.88914242669478378836924017799, 12.70294496311090280017295499420, 13.496667323331979598693551088405, 14.13932141129953302128453885499, 14.89777645872961871631324299130, 15.59693114397679807654899743847, 16.14370288477760199111820111031, 16.85126528911576009384782096299, 17.74714090695209153480782800089, 18.4681290084434301620728152733

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