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Properties

Label 1-4235-4235.2908-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.935 + 0.353i$
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  + (0.803 + 0.595i)2-s + (0.994 + 0.104i)3-s + (0.290 + 0.956i)4-s + (0.736 + 0.676i)6-s + (−0.336 + 0.941i)8-s + (0.978 + 0.207i)9-s + (0.189 + 0.981i)12-s + (−0.856 − 0.516i)13-s + (−0.830 + 0.556i)16-s + (0.353 − 0.935i)17-s + (0.662 + 0.749i)18-s + (0.548 − 0.836i)19-s + (0.618 − 0.786i)23-s + (−0.432 + 0.901i)24-s + (−0.380 − 0.924i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)2-s + (0.994 + 0.104i)3-s + (0.290 + 0.956i)4-s + (0.736 + 0.676i)6-s + (−0.336 + 0.941i)8-s + (0.978 + 0.207i)9-s + (0.189 + 0.981i)12-s + (−0.856 − 0.516i)13-s + (−0.830 + 0.556i)16-s + (0.353 − 0.935i)17-s + (0.662 + 0.749i)18-s + (0.548 − 0.836i)19-s + (0.618 − 0.786i)23-s + (−0.432 + 0.901i)24-s + (−0.380 − 0.924i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.935 + 0.353i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2908, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.935 + 0.353i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.205535429 + 0.7687627931i\)
\(L(\frac12)\) \(\approx\) \(4.205535429 + 0.7687627931i\)
\(L(1)\) \(\approx\) \(2.213735746 + 0.6562883692i\)
\(L(1)\) \(\approx\) \(2.213735746 + 0.6562883692i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.803 + 0.595i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.856 - 0.516i)T \)
17 \( 1 + (0.353 - 0.935i)T \)
19 \( 1 + (0.548 - 0.836i)T \)
23 \( 1 + (0.618 - 0.786i)T \)
29 \( 1 + (0.362 - 0.931i)T \)
31 \( 1 + (-0.123 - 0.992i)T \)
37 \( 1 + (0.572 - 0.820i)T \)
41 \( 1 + (0.870 - 0.491i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (-0.662 + 0.749i)T \)
53 \( 1 + (0.556 - 0.830i)T \)
59 \( 1 + (0.00951 + 0.999i)T \)
61 \( 1 + (0.398 - 0.917i)T \)
67 \( 1 + (0.0950 + 0.995i)T \)
71 \( 1 + (-0.254 - 0.967i)T \)
73 \( 1 + (0.846 + 0.532i)T \)
79 \( 1 + (-0.879 - 0.475i)T \)
83 \( 1 + (-0.170 + 0.985i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
97 \( 1 + (0.825 - 0.564i)T \)
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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

−18.58392669785376451139949496655, −17.9525486218622454403146822707, −16.82044938396817663243072694152, −16.10763217296073538410965501477, −15.34509510489449232304925615966, −14.544502877511363101777214299697, −14.457896286344546791339708567341, −13.57483415095665349974925479259, −12.90119677728464858232453640066, −12.36828121065334365115133806911, −11.70110271424592441626578675777, −10.793993279454950391824567164919, −9.99546284743896208696180113591, −9.599862907959241647347079286815, −8.75418590985914767909524085649, −7.89037295244315837115770332441, −7.11940965572325779362236075468, −6.46289491675383110852369622506, −5.47778046085458338843732416308, −4.73432519518120916402182590760, −3.98859512185894960505366550767, −3.232158654511051083063128631864, −2.73456386350795017069802177461, −1.63766339419943322312978872568, −1.296304301333038108375604930508, 0.7505050933612481076126496234, 2.41705353380967212682108679223, 2.54026169624499594582550172345, 3.46678147342882745677728326080, 4.28905227285836192896741584655, 4.91868454859255628865995087571, 5.618970392056751436277758095929, 6.68710744743117954627479231137, 7.350399930340542150734630711699, 7.77031458308408151890311769018, 8.60233288458091386544401730889, 9.33954119173921018284329809480, 9.961126383124245645135080259634, 11.00754506453063678406810006024, 11.772926384626204664833166779464, 12.582194879002563359110768941487, 13.11299209400878287530615146230, 13.789849219497611349965189957326, 14.38318724274869849754239864095, 14.95588882228715385296651223393, 15.54416282046214950705689346948, 16.15235755404186071896865027238, 16.866340684943475749651376266505, 17.67231055919184264847293181186, 18.33366479388077069685390172315

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