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Properties

Label 1-4235-4235.1249-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.0272 - 0.999i$
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  + (−0.948 − 0.318i)2-s + (0.669 − 0.743i)3-s + (0.797 + 0.603i)4-s + (−0.870 + 0.491i)6-s + (−0.564 − 0.825i)8-s + (−0.104 − 0.994i)9-s + (0.981 − 0.189i)12-s + (−0.974 − 0.226i)13-s + (0.272 + 0.962i)16-s + (0.625 − 0.780i)17-s + (−0.217 + 0.976i)18-s + (−0.935 + 0.353i)19-s + (0.786 + 0.618i)23-s + (−0.991 − 0.132i)24-s + (0.851 + 0.524i)26-s + (−0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (−0.948 − 0.318i)2-s + (0.669 − 0.743i)3-s + (0.797 + 0.603i)4-s + (−0.870 + 0.491i)6-s + (−0.564 − 0.825i)8-s + (−0.104 − 0.994i)9-s + (0.981 − 0.189i)12-s + (−0.974 − 0.226i)13-s + (0.272 + 0.962i)16-s + (0.625 − 0.780i)17-s + (−0.217 + 0.976i)18-s + (−0.935 + 0.353i)19-s + (0.786 + 0.618i)23-s + (−0.991 − 0.132i)24-s + (0.851 + 0.524i)26-s + (−0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0272 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0272 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.0272 - 0.999i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.0272 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9314467994 - 0.9572289633i\)
\(L(\frac12)\) \(\approx\) \(0.9314467994 - 0.9572289633i\)
\(L(1)\) \(\approx\) \(0.8020102574 - 0.3674195513i\)
\(L(1)\) \(\approx\) \(0.8020102574 - 0.3674195513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.948 - 0.318i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (-0.974 - 0.226i)T \)
17 \( 1 + (0.625 - 0.780i)T \)
19 \( 1 + (-0.935 + 0.353i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (0.254 + 0.967i)T \)
31 \( 1 + (0.905 - 0.424i)T \)
37 \( 1 + (0.999 - 0.0190i)T \)
41 \( 1 + (0.993 + 0.113i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.217 - 0.976i)T \)
53 \( 1 + (-0.272 + 0.962i)T \)
59 \( 1 + (0.595 + 0.803i)T \)
61 \( 1 + (0.749 + 0.662i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
71 \( 1 + (-0.998 - 0.0570i)T \)
73 \( 1 + (0.0665 + 0.997i)T \)
79 \( 1 + (0.179 - 0.983i)T \)
83 \( 1 + (0.466 - 0.884i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
97 \( 1 + (0.610 - 0.791i)T \)
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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

−18.77350737443043929484422205808, −17.656480681609128871840563396761, −17.11246086279903662699682447686, −16.611593624717807212821752139749, −15.87161882424900802149479999116, −15.17972930256930820347124215662, −14.67200349566790010429977767421, −14.22817657745413122827605444959, −13.139096216150875452821823289766, −12.36653285459975985174441580874, −11.38336176095764880356304857453, −10.80684624211941645797551307663, −9.99330467229355987263342972234, −9.70351208456269266331187631978, −8.8347727833404632626086315383, −8.22828924706222645721320971608, −7.72257818710440423003221650638, −6.778025375004267320749305263468, −6.11226656523899225143541183432, −5.06103867151923989364208349480, −4.51551096536305635709836920311, −3.44596417717886491472691253847, −2.5195199810673088162956867818, −2.07052618545209886716399678771, −0.80706411314184814160561359865, 0.60910091757645823773578440131, 1.37687829599265000145247619467, 2.32722939593686260952341564586, 2.84251276346498419952856182375, 3.58230537087859254731922348604, 4.661348300148011237479331329839, 5.8087445733029054119066311189, 6.62488423846608108431269912643, 7.33744405382482752852009277706, 7.746570629605189789250740429317, 8.5528651397976423525556378331, 9.152116880713554604561271631789, 9.861675660877408405717399505990, 10.426211861494995412803209837083, 11.57153469525494443504635755759, 11.8814005587392049256438697738, 12.824783375469893206068259156154, 13.11652250149800907277390850741, 14.2564151092585603741499949910, 14.82305723905573088503089123159, 15.4553195590207651903874756697, 16.412176116880595460036622644251, 17.02304774709981110213343088932, 17.67358571262172959316140948013, 18.29880593103381366631947658120

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