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Properties

Label 1-4235-4235.172-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.299 - 0.954i$
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  + (0.475 − 0.879i)2-s + (0.994 + 0.104i)3-s + (−0.548 − 0.836i)4-s + (0.564 − 0.825i)6-s + (−0.996 + 0.0855i)8-s + (0.978 + 0.207i)9-s + (−0.458 − 0.888i)12-s + (−0.967 − 0.254i)13-s + (−0.398 + 0.917i)16-s + (0.938 − 0.345i)17-s + (0.647 − 0.761i)18-s + (0.272 + 0.962i)19-s + (−0.0950 + 0.995i)23-s + (−0.999 − 0.0190i)24-s + (−0.683 + 0.730i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.475 − 0.879i)2-s + (0.994 + 0.104i)3-s + (−0.548 − 0.836i)4-s + (0.564 − 0.825i)6-s + (−0.996 + 0.0855i)8-s + (0.978 + 0.207i)9-s + (−0.458 − 0.888i)12-s + (−0.967 − 0.254i)13-s + (−0.398 + 0.917i)16-s + (0.938 − 0.345i)17-s + (0.647 − 0.761i)18-s + (0.272 + 0.962i)19-s + (−0.0950 + 0.995i)23-s + (−0.999 − 0.0190i)24-s + (−0.683 + 0.730i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.299 - 0.954i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.299 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.772256557 - 2.414149949i\)
\(L(\frac12)\) \(\approx\) \(1.772256557 - 2.414149949i\)
\(L(1)\) \(\approx\) \(1.490657668 - 0.9039899640i\)
\(L(1)\) \(\approx\) \(1.490657668 - 0.9039899640i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.475 - 0.879i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.967 - 0.254i)T \)
17 \( 1 + (0.938 - 0.345i)T \)
19 \( 1 + (0.272 + 0.962i)T \)
23 \( 1 + (-0.0950 + 0.995i)T \)
29 \( 1 + (-0.466 - 0.884i)T \)
31 \( 1 + (0.161 - 0.986i)T \)
37 \( 1 + (0.780 - 0.625i)T \)
41 \( 1 + (-0.610 - 0.791i)T \)
43 \( 1 + (0.755 - 0.654i)T \)
47 \( 1 + (-0.647 - 0.761i)T \)
53 \( 1 + (-0.917 + 0.398i)T \)
59 \( 1 + (0.991 + 0.132i)T \)
61 \( 1 + (0.851 + 0.524i)T \)
67 \( 1 + (-0.971 + 0.235i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (-0.999 + 0.00951i)T \)
79 \( 1 + (0.797 - 0.603i)T \)
83 \( 1 + (0.676 - 0.736i)T \)
89 \( 1 + (-0.928 - 0.371i)T \)
97 \( 1 + (0.856 + 0.516i)T \)
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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

−18.44544629469111700649380273833, −17.92178594995772647090079072117, −17.068081107704134645431863732162, −16.37495092430941671633847653315, −15.832592955458081554576777470980, −14.9232506854466245582672095556, −14.57029460186168971857331302396, −14.095508467642448512314354865068, −13.22195767062265855941225778021, −12.66931762175984615815017964832, −12.144612176832791744563616682528, −11.1261756836518918796614088330, −9.95617777375071511511342305360, −9.50800115575944950453060425106, −8.68127871064032203716446608856, −8.10362191385575255404790092654, −7.41721794353983238217075496247, −6.82783756844221571093490043363, −6.12865310059612306802327634551, −4.866508612100576837262676450, −4.713831428093571345352158299108, −3.548295422994158022975285442705, −3.019732523342934023837429765645, −2.23066354123878258575735060275, −1.00889131776447776173102528503, 0.67360930795560853742813598879, 1.76591119291709648196742113347, 2.32041089909543518762576400424, 3.1819447502771219180118297086, 3.74847315357883197076474512649, 4.46361604754883872055775522410, 5.37608426795949529840054764868, 5.944597092917631401151219244081, 7.27551655780962190367049056763, 7.741919324525514338126365516913, 8.62181037305333227279656502421, 9.54598051121301594121780479775, 9.83253936325601560295623503824, 10.4375612288376768064988644754, 11.52298807557146049855981867445, 12.05179645057533282057833364439, 12.80050805397208910010876712374, 13.40551863138747660095886851857, 14.09025314879088050143299608288, 14.6269656190376796590238142586, 15.16206442120453012209686300794, 15.90731451061149762545104525258, 16.83040692332843372216552129636, 17.69652836966650711016364130022, 18.51478991861736665166214189286

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