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Properties

Label 1-4235-4235.1179-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.970 - 0.240i$
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  + (0.749 + 0.662i)2-s + (−0.978 + 0.207i)3-s + (0.123 + 0.992i)4-s + (−0.870 − 0.491i)6-s + (−0.564 + 0.825i)8-s + (0.913 − 0.406i)9-s + (−0.327 − 0.945i)12-s + (−0.974 + 0.226i)13-s + (−0.969 + 0.244i)16-s + (−0.988 + 0.151i)17-s + (0.953 + 0.299i)18-s + (0.161 + 0.986i)19-s + (−0.928 − 0.371i)23-s + (0.380 − 0.924i)24-s + (−0.879 − 0.475i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.749 + 0.662i)2-s + (−0.978 + 0.207i)3-s + (0.123 + 0.992i)4-s + (−0.870 − 0.491i)6-s + (−0.564 + 0.825i)8-s + (0.913 − 0.406i)9-s + (−0.327 − 0.945i)12-s + (−0.974 + 0.226i)13-s + (−0.969 + 0.244i)16-s + (−0.988 + 0.151i)17-s + (0.953 + 0.299i)18-s + (0.161 + 0.986i)19-s + (−0.928 − 0.371i)23-s + (0.380 − 0.924i)24-s + (−0.879 − 0.475i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.970 - 0.240i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.970 - 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6442594275 - 0.07859889582i\)
\(L(\frac12)\) \(\approx\) \(0.6442594275 - 0.07859889582i\)
\(L(1)\) \(\approx\) \(0.7859668221 + 0.4418261616i\)
\(L(1)\) \(\approx\) \(0.7859668221 + 0.4418261616i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.749 + 0.662i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.974 + 0.226i)T \)
17 \( 1 + (-0.988 + 0.151i)T \)
19 \( 1 + (0.161 + 0.986i)T \)
23 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (0.254 - 0.967i)T \)
31 \( 1 + (-0.820 + 0.572i)T \)
37 \( 1 + (-0.483 - 0.875i)T \)
41 \( 1 + (0.993 - 0.113i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.953 - 0.299i)T \)
53 \( 1 + (0.969 + 0.244i)T \)
59 \( 1 + (0.398 + 0.917i)T \)
61 \( 1 + (-0.948 - 0.318i)T \)
67 \( 1 + (-0.580 + 0.814i)T \)
71 \( 1 + (-0.998 + 0.0570i)T \)
73 \( 1 + (0.830 + 0.556i)T \)
79 \( 1 + (0.761 - 0.647i)T \)
83 \( 1 + (0.466 + 0.884i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
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

−18.212448265050448187455219698517, −17.96873081840450054786212743913, −17.10627830028112869519050706348, −16.307083059003505954217699746864, −15.58216425628039239096520811343, −15.06904870892751247151590982171, −14.16683231376044732054894500006, −13.387025260819727103386017198966, −12.95027939447303470123531075475, −12.11445785928070849157740461071, −11.74347469658547983396684191152, −10.94006796812512357935426067002, −10.47940532448745127795093791240, −9.65394031090771563204839632092, −9.03554107042989426145188720919, −7.7187188517692742037061677652, −6.983133162606201090344555787574, −6.39160770112385328432989912543, −5.57239352027754164880453452084, −4.910689583223950114143070425322, −4.43954237529071698574498221610, −3.472790879934870031533218930242, −2.46443459761755609986161231901, −1.84035962177754581261512182596, −0.78811916417449991109443124477, 0.1927295653176080289992100290, 1.78906175952410263190129429961, 2.56999954996122489364032215177, 3.86035741924943463751570581074, 4.19886605659969398318285191694, 5.0381764663416772014470622934, 5.71782575019877897762862527180, 6.2678825838585264045910030265, 7.09137730459076337868652730327, 7.56940845943548413384653837785, 8.56182758434348733122367927570, 9.36483688112317644609874699917, 10.2610612961952298797102159643, 10.888109409395160997827420199185, 11.91303208789479684353247387195, 12.091108312433038107819288267490, 12.84599732053485218987867805234, 13.62256089596993701633194280834, 14.36554668309287187732169217844, 15.04982294022390898659530235837, 15.67156975141042409402271101261, 16.39898472536047698820792356246, 16.74927877333174994647087079055, 17.62902889247487121827716832447, 17.928919492928207208635274296032

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