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Properties

Label 1-4235-4235.139-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.236 + 0.971i$
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  + (0.774 + 0.633i)2-s + (−0.809 − 0.587i)3-s + (0.198 + 0.980i)4-s + (−0.254 − 0.967i)6-s + (−0.466 + 0.884i)8-s + (0.309 + 0.951i)9-s + (0.415 − 0.909i)12-s + (−0.993 − 0.113i)13-s + (−0.921 + 0.389i)16-s + (0.564 − 0.825i)17-s + (−0.362 + 0.931i)18-s + (0.941 + 0.336i)19-s + (0.654 + 0.755i)23-s + (0.897 − 0.441i)24-s + (−0.696 − 0.717i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (0.774 + 0.633i)2-s + (−0.809 − 0.587i)3-s + (0.198 + 0.980i)4-s + (−0.254 − 0.967i)6-s + (−0.466 + 0.884i)8-s + (0.309 + 0.951i)9-s + (0.415 − 0.909i)12-s + (−0.993 − 0.113i)13-s + (−0.921 + 0.389i)16-s + (0.564 − 0.825i)17-s + (−0.362 + 0.931i)18-s + (0.941 + 0.336i)19-s + (0.654 + 0.755i)23-s + (0.897 − 0.441i)24-s + (−0.696 − 0.717i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.236 + 0.971i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.236 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.476115400 + 1.159796548i\)
\(L(\frac12)\) \(\approx\) \(1.476115400 + 1.159796548i\)
\(L(1)\) \(\approx\) \(1.165254011 + 0.4055090989i\)
\(L(1)\) \(\approx\) \(1.165254011 + 0.4055090989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.774 + 0.633i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.993 - 0.113i)T \)
17 \( 1 + (0.564 - 0.825i)T \)
19 \( 1 + (0.941 + 0.336i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.610 + 0.791i)T \)
31 \( 1 + (0.736 - 0.676i)T \)
37 \( 1 + (0.870 + 0.491i)T \)
41 \( 1 + (-0.998 - 0.0570i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.362 - 0.931i)T \)
53 \( 1 + (0.921 + 0.389i)T \)
59 \( 1 + (0.998 - 0.0570i)T \)
61 \( 1 + (0.774 - 0.633i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (-0.974 - 0.226i)T \)
79 \( 1 + (0.985 - 0.170i)T \)
83 \( 1 + (-0.516 - 0.856i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.897 - 0.441i)T \)
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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

−18.18608565304657738602152166313, −17.60614447186670037159977370935, −16.69481588965535993949453405259, −16.25830262914063887043847754244, −15.3002744864087559760159631979, −14.86838333729547864339374688380, −14.279601162927952022553180417704, −13.2701395620786660848836636039, −12.68174939464730404021840085375, −11.97381928457649957359136619001, −11.52876811844331467030635409536, −10.78369924244764388270109054214, −10.05423450564155808189635737870, −9.695359487865770872586936281221, −8.82697117956392623481642231468, −7.6084898956585564742801376401, −6.76391119797115006325386265770, −6.10042839452546682109072295063, −5.346896322800719769740225259790, −4.78574099348964325592115110366, −4.14073592319856321031061240313, −3.29135053609669211953381066329, −2.5764516833865747770083842212, −1.45481364849773740598916778090, −0.58802672924962205115049486224, 0.81308863957329115688910201717, 1.96904810069335976032368686260, 2.82676067206117729978410905730, 3.6353079475471602890529371900, 4.722708853710550732370687080945, 5.27239416185410036608951458873, 5.69274676186721440239337939713, 6.701108664451937816079003403186, 7.29366097087675069405344512852, 7.650968123094926706589179061436, 8.5881572499260398701781688884, 9.627339269654891697316665358008, 10.33134163389526106528794308737, 11.52731323375868367124575645923, 11.72350175300013395838030045772, 12.36535311738084146864060461406, 13.2751663781282259975679493149, 13.54070777498001318719689624929, 14.4851394815599485296310365346, 15.0562245918739907250066562824, 15.94303968152913410506603400311, 16.490445567142258291998475839594, 17.08855646035573000311413888030, 17.60260795159846993826160600922, 18.42471906827123146396110754742

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