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Properties

Label 1-4235-4235.2007-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.242 - 0.970i$
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  + (0.132 − 0.991i)2-s + (0.207 − 0.978i)3-s + (−0.964 − 0.263i)4-s + (−0.941 − 0.336i)6-s + (−0.389 + 0.921i)8-s + (−0.913 − 0.406i)9-s + (−0.458 + 0.888i)12-s + (−0.931 − 0.362i)13-s + (0.861 + 0.508i)16-s + (0.962 − 0.272i)17-s + (−0.524 + 0.851i)18-s + (−0.830 − 0.556i)19-s + (0.0950 + 0.995i)23-s + (0.820 + 0.572i)24-s + (−0.483 + 0.875i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.132 − 0.991i)2-s + (0.207 − 0.978i)3-s + (−0.964 − 0.263i)4-s + (−0.941 − 0.336i)6-s + (−0.389 + 0.921i)8-s + (−0.913 − 0.406i)9-s + (−0.458 + 0.888i)12-s + (−0.931 − 0.362i)13-s + (0.861 + 0.508i)16-s + (0.962 − 0.272i)17-s + (−0.524 + 0.851i)18-s + (−0.830 − 0.556i)19-s + (0.0950 + 0.995i)23-s + (0.820 + 0.572i)24-s + (−0.483 + 0.875i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.242 - 0.970i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.242 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8699863804 - 1.114015186i\)
\(L(\frac12)\) \(\approx\) \(0.8699863804 - 1.114015186i\)
\(L(1)\) \(\approx\) \(0.7175505736 - 0.7042594990i\)
\(L(1)\) \(\approx\) \(0.7175505736 - 0.7042594990i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.132 - 0.991i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (-0.931 - 0.362i)T \)
17 \( 1 + (0.962 - 0.272i)T \)
19 \( 1 + (-0.830 - 0.556i)T \)
23 \( 1 + (0.0950 + 0.995i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (-0.449 + 0.893i)T \)
37 \( 1 + (0.353 - 0.935i)T \)
41 \( 1 + (0.564 + 0.825i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (0.524 + 0.851i)T \)
53 \( 1 + (-0.508 - 0.861i)T \)
59 \( 1 + (-0.432 - 0.901i)T \)
61 \( 1 + (-0.380 + 0.924i)T \)
67 \( 1 + (0.971 + 0.235i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (0.299 + 0.953i)T \)
79 \( 1 + (0.290 + 0.956i)T \)
83 \( 1 + (0.980 + 0.198i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
97 \( 1 + (-0.996 - 0.0855i)T \)
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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

−18.648838206324254868170421668073, −17.50537224358284641116967879420, −16.91559497767851743017135556660, −16.63595832735538609474845917863, −15.84750289886855840067762508839, −15.09963890341182142602977235040, −14.67645397565564144657137555533, −14.13316447066734010455602183867, −13.430912607377478452994799452096, −12.340763777130626806757295065009, −12.04694841487775675653122501453, −10.667291878443548308795923288162, −10.27164088068772293608168336603, −9.426201408017408666839172786196, −8.94124081950833948216062505477, −8.0607399580743723778630472612, −7.6348525703884231931374467211, −6.54121083073738702495446607989, −5.93390931159249018142680080952, −5.13880401881015600786439821818, −4.47338158121054308349404047967, −3.9275019906381208499051718317, −3.04109028551152701519328414975, −2.13774451251537634045090093970, −0.53492712769257187936418196461, 0.72361462458076995980817896240, 1.42463639639413149963745446275, 2.41160113468881140911590339382, 2.86761546432811990635086934103, 3.68550912419097437424236717793, 4.72229416830865470320899342740, 5.416721771984834625864747259501, 6.16894422882444893720718387539, 7.16475250866494831923884746062, 7.81537762285882388146586150697, 8.512172805457104844062450501907, 9.34122011067186787484742716046, 9.86420784035037086144569965782, 10.86559710564027114480599702342, 11.37733834238139787222284439049, 12.26666696185293030589847611294, 12.609356314218228056780235010001, 13.16986039351181638431157692027, 14.16594982695718187520759620962, 14.35740091143449556441764570725, 15.18370459609218261461182325458, 16.25360368878670549950947833600, 17.30379280239633634788561535897, 17.61498681399242424766462114243, 18.25200482124429886658297541917

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