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Properties

Label 1-4235-4235.1033-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.844 - 0.534i$
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  + (−0.458 − 0.888i)2-s + (0.866 − 0.5i)3-s + (−0.580 + 0.814i)4-s + (−0.841 − 0.540i)6-s + (0.989 + 0.142i)8-s + (0.5 − 0.866i)9-s + (−0.0950 + 0.995i)12-s + (0.909 − 0.415i)13-s + (−0.327 − 0.945i)16-s + (−0.690 + 0.723i)17-s + (−0.998 − 0.0475i)18-s + (0.723 − 0.690i)19-s + (0.945 − 0.327i)23-s + (0.928 − 0.371i)24-s + (−0.786 − 0.618i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.458 − 0.888i)2-s + (0.866 − 0.5i)3-s + (−0.580 + 0.814i)4-s + (−0.841 − 0.540i)6-s + (0.989 + 0.142i)8-s + (0.5 − 0.866i)9-s + (−0.0950 + 0.995i)12-s + (0.909 − 0.415i)13-s + (−0.327 − 0.945i)16-s + (−0.690 + 0.723i)17-s + (−0.998 − 0.0475i)18-s + (0.723 − 0.690i)19-s + (0.945 − 0.327i)23-s + (0.928 − 0.371i)24-s + (−0.786 − 0.618i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.844 - 0.534i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.844 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4981304838 - 1.718082174i\)
\(L(\frac12)\) \(\approx\) \(0.4981304838 - 1.718082174i\)
\(L(1)\) \(\approx\) \(0.8916427341 - 0.7072233201i\)
\(L(1)\) \(\approx\) \(0.8916427341 - 0.7072233201i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.458 - 0.888i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.909 - 0.415i)T \)
17 \( 1 + (-0.690 + 0.723i)T \)
19 \( 1 + (0.723 - 0.690i)T \)
23 \( 1 + (0.945 - 0.327i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.995 + 0.0950i)T \)
37 \( 1 + (0.814 - 0.580i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (0.998 - 0.0475i)T \)
53 \( 1 + (-0.945 - 0.327i)T \)
59 \( 1 + (0.888 + 0.458i)T \)
61 \( 1 + (-0.0475 - 0.998i)T \)
67 \( 1 + (-0.998 - 0.0475i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.189 - 0.981i)T \)
79 \( 1 + (0.928 + 0.371i)T \)
83 \( 1 + (-0.755 + 0.654i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
97 \( 1 + (0.989 + 0.142i)T \)
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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

−18.61658490535281439690115629024, −18.11424386546269017555032623180, −17.102657760708388684600999886, −16.45467381798761555130464670952, −15.935679143046874331179779958, −15.37987422714161907049640109746, −14.63352208001352370926439707622, −14.15372993449581094790969160045, −13.35517561245296635735248677058, −13.02689340109181751519463198313, −11.565643662620469581725912837801, −10.931046141911897261145689783634, −10.19300276426657386024159336003, −9.31309769186884617311165858397, −9.0846561369379558634690632884, −8.34032162385661011764798952398, −7.48361575239625033052745471234, −7.10706988078751369711438497077, −6.04980337993257739125036124357, −5.33820147491919363842173935427, −4.535978879473636179741202695067, −3.843685860628189409056311958497, −2.999943853142476413461984626935, −1.88281009823884719507089473328, −1.12677777380575806818284720161, 0.55892228709109527790136665764, 1.4188951684205593638897515024, 2.16076537973484261912607680088, 2.923744263229417916580593336248, 3.63352938982606947092339263160, 4.22123288270136712078727093114, 5.31160009719112193182972750559, 6.35232477211918581910926060920, 7.26482754704395246523973100742, 7.76769297649843712596716073556, 8.650583975557288826791383259011, 9.04457736579973015669102245719, 9.64736508713738304644186280088, 10.7444351615994263241555387236, 11.04727616444718907670552587166, 11.998379060383004672864076434241, 12.80744925298386448258257279052, 13.17694753769916684973196066495, 13.74138005140285663758079162280, 14.60406346171801842899291959295, 15.33324514072100545948138961008, 16.07118633821692055840587940715, 16.99851348422592094979401930834, 17.72675117765600914690547115577, 18.240427153696571536423790584787

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