↑

Properties

Label 1-4235-4235.1173-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.366 + 0.930i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.976 − 0.217i)2-s + (−0.994 − 0.104i)3-s + (0.905 − 0.424i)4-s + (−0.993 + 0.113i)6-s + (0.791 − 0.610i)8-s + (0.978 + 0.207i)9-s + (−0.945 + 0.327i)12-s + (−0.389 + 0.921i)13-s + (0.640 − 0.768i)16-s + (−0.703 − 0.710i)17-s + (0.999 − 0.00951i)18-s + (0.988 + 0.151i)19-s + (−0.371 + 0.928i)23-s + (−0.851 + 0.524i)24-s + (−0.179 + 0.983i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.976 − 0.217i)2-s + (−0.994 − 0.104i)3-s + (0.905 − 0.424i)4-s + (−0.993 + 0.113i)6-s + (0.791 − 0.610i)8-s + (0.978 + 0.207i)9-s + (−0.945 + 0.327i)12-s + (−0.389 + 0.921i)13-s + (0.640 − 0.768i)16-s + (−0.703 − 0.710i)17-s + (0.999 − 0.00951i)18-s + (0.988 + 0.151i)19-s + (−0.371 + 0.928i)23-s + (−0.851 + 0.524i)24-s + (−0.179 + 0.983i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.366 + 0.930i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.366 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5162403100 + 0.7578853661i\)
\(L(\frac12)\) \(\approx\) \(0.5162403100 + 0.7578853661i\)
\(L(1)\) \(\approx\) \(1.197743487 - 0.07585134014i\)
\(L(1)\) \(\approx\) \(1.197743487 - 0.07585134014i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.976 - 0.217i)T \)
3 \( 1 + (-0.994 - 0.104i)T \)
13 \( 1 + (-0.389 + 0.921i)T \)
17 \( 1 + (-0.703 - 0.710i)T \)
19 \( 1 + (0.988 + 0.151i)T \)
23 \( 1 + (-0.371 + 0.928i)T \)
29 \( 1 + (-0.998 + 0.0570i)T \)
31 \( 1 + (-0.999 - 0.0190i)T \)
37 \( 1 + (0.730 + 0.683i)T \)
41 \( 1 + (-0.198 + 0.980i)T \)
43 \( 1 + (-0.281 + 0.959i)T \)
47 \( 1 + (-0.999 - 0.00951i)T \)
53 \( 1 + (-0.768 + 0.640i)T \)
59 \( 1 + (-0.749 + 0.662i)T \)
61 \( 1 + (-0.953 + 0.299i)T \)
67 \( 1 + (-0.814 - 0.580i)T \)
71 \( 1 + (0.774 - 0.633i)T \)
73 \( 1 + (-0.962 - 0.272i)T \)
79 \( 1 + (0.997 - 0.0760i)T \)
83 \( 1 + (0.441 - 0.897i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
97 \( 1 + (-0.999 + 0.0285i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.91893758148065409051708695879, −17.402637465095844286474140817, −16.62786047631756029978395486336, −16.16796432048790298861082281794, −15.35713576936821775209360332777, −14.963912021228079477704686687201, −14.08254978346807459191620015714, −13.229915009450462994863039464105, −12.65851522709649640015101414070, −12.24127597105796530965054853641, −11.311906442836230296425995592582, −10.88601546270512432688695511772, −10.20633089883588802140503400449, −9.31724099371375994100412765913, −8.179931302343574020878237087293, −7.457300782918980292951031062806, −6.82356029722640944094257731778, −6.04724136539587859692885961388, −5.45926772081385958243878344102, −4.89759078427810115865794162005, −4.043371416198400419391337497318, −3.42130277748250380864906956668, −2.33656889040947871256594402893, −1.52706209600405063356588313736, −0.18624153864377840116105097822, 1.318916998780223826649995403575, 1.83205416908735881336819586327, 2.93665745583372718096169881081, 3.77115594496974168293538820602, 4.71703102203591287861765586482, 4.9674508742677205663343978501, 5.9957191289022497721721646241, 6.39401289021393184203641018552, 7.38287097711210389420482659347, 7.63216816553007548193980279844, 9.34775619765851057710875541641, 9.663929215999977299038376884262, 10.669242511258928544736767650531, 11.41058709725429610226882450348, 11.65195222698857626027783437685, 12.34975376952571821891204635452, 13.25443464866665396852830981058, 13.55860224812104545321743504808, 14.489522349212678527080243756138, 15.1744826656964835435625102209, 15.94476090948233816484226124423, 16.47414304489634874979900041682, 16.95852045351651679317430065668, 18.10672688997278624538910849686, 18.39466636880304066310669172774

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