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Properties

Label 1-4235-4235.1924-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.355 - 0.934i$
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  + (0.415 − 0.909i)2-s + 3-s + (−0.654 − 0.755i)4-s + (0.415 − 0.909i)6-s + (−0.959 + 0.281i)8-s + 9-s + (−0.654 − 0.755i)12-s + (0.654 + 0.755i)13-s + (−0.142 + 0.989i)16-s + (−0.841 − 0.540i)17-s + (0.415 − 0.909i)18-s + (0.841 − 0.540i)19-s + (0.142 − 0.989i)23-s + (−0.959 + 0.281i)24-s + (0.959 − 0.281i)26-s + 27-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + 3-s + (−0.654 − 0.755i)4-s + (0.415 − 0.909i)6-s + (−0.959 + 0.281i)8-s + 9-s + (−0.654 − 0.755i)12-s + (0.654 + 0.755i)13-s + (−0.142 + 0.989i)16-s + (−0.841 − 0.540i)17-s + (0.415 − 0.909i)18-s + (0.841 − 0.540i)19-s + (0.142 − 0.989i)23-s + (−0.959 + 0.281i)24-s + (0.959 − 0.281i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.355 - 0.934i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1924, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.355 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.770551360 - 2.567827983i\)
\(L(\frac12)\) \(\approx\) \(1.770551360 - 2.567827983i\)
\(L(1)\) \(\approx\) \(1.476306301 - 0.9748860590i\)
\(L(1)\) \(\approx\) \(1.476306301 - 0.9748860590i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.415 - 0.909i)T \)
3 \( 1 + T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (0.654 - 0.755i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.415 + 0.909i)T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−18.32678617823703647109950830298, −18.02426579570321382583831377564, −17.1228410097803319235484807103, −16.37205316313382851202725669578, −15.57499378114300435922115073595, −15.28519683882716077060070804397, −14.63481101643669347993838979525, −13.748426434386042082133078167012, −13.40594482886121447728009000102, −12.841294544207148105811447507378, −11.98007325943320682847808348600, −11.10571632092271605764296223221, −9.9964041942360424321479287379, −9.51886867034319751216873688547, −8.579051287094477643942940161989, −8.19472030626634035585862842306, −7.52526700608825643063841301971, −6.78005246077072735136991271304, −6.02132621674468948052887718908, −5.21817427901835346611947971339, −4.4034929619336213345888735770, −3.54960162257463678968900666030, −3.20417913250785902849136905613, −2.10599143941794523651183978426, −1.04644849622576655388524139695, 0.72526874094897149654240897718, 1.65224589351299386944145586315, 2.43737515429684448838678088534, 2.97191839887029868153284775330, 3.94302335181820137958124726885, 4.37050737430529162792795388175, 5.20513969325445548262281255452, 6.250714890915026759963237819665, 6.976626625658195979725019347730, 7.883512134743032138899631912559, 8.84764323023703370493958770955, 9.17647215920600828150155915751, 9.79986127515822855327734093039, 10.783456325168112665518804431561, 11.23369737070838281600080050326, 12.10092896590585360158794440801, 12.83418604944463046623979196342, 13.49481548451742417827601800911, 13.91238341543434452051082754394, 14.56137366194571707019925071026, 15.30417255835876024386911879344, 15.889691913846083771224453806616, 16.73978264172328981786167991892, 17.969335234861241847229576617503, 18.30307623197400605497489536690

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