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Properties

Label 1-4235-4235.1108-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: $\chi_{4235} (1108, \cdot )$
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

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

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