↑

Properties

Label 2-3536-884.387-c0-0-0
Degree $2$
Conductor $3536$
Sign $0.988 + 0.151i$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
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  + (−0.366 + 0.366i)5-s + (−0.866 − 0.5i)9-s + (0.866 − 0.5i)13-s + 17-s + 0.732i·25-s + (−0.133 + 0.5i)29-s + (0.5 + 0.133i)37-s + (0.5 − 1.86i)41-s + (0.5 − 0.133i)45-s + (0.866 − 0.5i)49-s − 1.73i·53-s + (0.5 + 1.86i)61-s + (−0.133 + 0.5i)65-s + (1.36 − 1.36i)73-s + (0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (−0.366 + 0.366i)5-s + (−0.866 − 0.5i)9-s + (0.866 − 0.5i)13-s + 17-s + 0.732i·25-s + (−0.133 + 0.5i)29-s + (0.5 + 0.133i)37-s + (0.5 − 1.86i)41-s + (0.5 − 0.133i)45-s + (0.866 − 0.5i)49-s − 1.73i·53-s + (0.5 + 1.86i)61-s + (−0.133 + 0.5i)65-s + (1.36 − 1.36i)73-s + (0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $0.988 + 0.151i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (3039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 0.988 + 0.151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166220991\)
\(L(\frac12)\) \(\approx\) \(1.166220991\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 - T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.771888885915433398474582630943, −7.959358249013089719617686398731, −7.34013416773516941290140877007, −6.45583338762518527941542073244, −5.71868893220346812891191458781, −5.13432974125040069766871913257, −3.66804113452227572108550084597, −3.49258317388776706089010304014, −2.34847499337767040773169263661, −0.900200859651438064744022719550, 1.03244166150074404073842159449, 2.32948205132109371842667147943, 3.27059105135473391541348153108, 4.17560127072179052753030283958, 4.93067871624412781922675743816, 5.87158290160693019701617679952, 6.33750159588739553629024152575, 7.51858170722912701545522669131, 8.072288035373869801776668319456, 8.630280283475961326767266953629

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