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Properties

Label 1-4235-4235.1224-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.391 + 0.920i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.897 − 0.441i)2-s + (−0.809 + 0.587i)3-s + (0.610 + 0.791i)4-s + (0.985 − 0.170i)6-s + (−0.198 − 0.980i)8-s + (0.309 − 0.951i)9-s + (−0.959 − 0.281i)12-s + (−0.564 + 0.825i)13-s + (−0.254 + 0.967i)16-s + (−0.921 − 0.389i)17-s + (−0.696 + 0.717i)18-s + (−0.974 − 0.226i)19-s + (−0.841 − 0.540i)23-s + (0.736 + 0.676i)24-s + (0.870 − 0.491i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.897 − 0.441i)2-s + (−0.809 + 0.587i)3-s + (0.610 + 0.791i)4-s + (0.985 − 0.170i)6-s + (−0.198 − 0.980i)8-s + (0.309 − 0.951i)9-s + (−0.959 − 0.281i)12-s + (−0.564 + 0.825i)13-s + (−0.254 + 0.967i)16-s + (−0.921 − 0.389i)17-s + (−0.696 + 0.717i)18-s + (−0.974 − 0.226i)19-s + (−0.841 − 0.540i)23-s + (0.736 + 0.676i)24-s + (0.870 − 0.491i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.391 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1321122475 + 0.1998189397i\)
\(L(\frac12)\) \(\approx\) \(0.1321122475 + 0.1998189397i\)
\(L(1)\) \(\approx\) \(0.4515959501 + 0.003880167628i\)
\(L(1)\) \(\approx\) \(0.4515959501 + 0.003880167628i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.897 - 0.441i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.564 + 0.825i)T \)
17 \( 1 + (-0.921 - 0.389i)T \)
19 \( 1 + (-0.974 - 0.226i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (0.0855 + 0.996i)T \)
31 \( 1 + (0.0285 - 0.999i)T \)
37 \( 1 + (-0.941 - 0.336i)T \)
41 \( 1 + (0.466 + 0.884i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.696 + 0.717i)T \)
53 \( 1 + (0.254 + 0.967i)T \)
59 \( 1 + (0.466 - 0.884i)T \)
61 \( 1 + (-0.897 + 0.441i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (-0.362 - 0.931i)T \)
79 \( 1 + (0.993 - 0.113i)T \)
83 \( 1 + (0.774 + 0.633i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.736 - 0.676i)T \)
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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

−17.759195484088197353889865279202, −17.40312056305562733722193861117, −16.96036076063719098256839920980, −16.00525130984069615018748377293, −15.54086036902654577356814515655, −14.83660349081951082608593483739, −13.89681430708934027650744596206, −13.23760985782805592263020450045, −12.30389331046805966946041891716, −11.85437958283605535751356196985, −10.94391331674632801971616942438, −10.399273182069151121219916478840, −9.93300195287828370579031254889, −8.75149316333253834497742840509, −8.27834253013063435002146534280, −7.46808358818684706331159252185, −6.88244232623377348822168344738, −6.19832542735693410693829911110, −5.543493115173714648292391086161, −4.88775037925035575675565950117, −3.81093854069529210095021719332, −2.34818566683117553559834536072, −1.98783418234229357358303368117, −0.89036890201756206916664302752, −0.110376214368095599669907375128, 0.50846897363996043653983623437, 1.68343671225410107054814678778, 2.40321841130083853908646040681, 3.396237762185558271076812921082, 4.35808416611760649944536042367, 4.699213916656208925247534377966, 6.043683799253115032901001891646, 6.546444265950939187791059473998, 7.24123285863411324319644586929, 8.153453959085506899020507609586, 9.12771980810592713250526720176, 9.35428078643397221079703163641, 10.27799657736330579362284109234, 10.856166875681518035591994985351, 11.3557985479028650617411926721, 12.15619131381993395450497299555, 12.56439830215327799991880793540, 13.51382042582652902711207814782, 14.55419427103563815187497588457, 15.29655564310041723700044125660, 15.945697639064117450308430345793, 16.632581269008939023068788359041, 16.98031959402331571662497910692, 17.839418268221259149351465007633, 18.20160421036018454136220758195

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