↑

Properties

Label 2-3900-195.158-c0-0-1
Degree $2$
Conductor $3900$
Sign $0.460 - 0.887i$
Analytic cond. $1.94635$
Root an. cond. $1.39511$
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  + (0.258 − 0.965i)3-s + (−0.707 + 1.22i)7-s + (−0.866 − 0.499i)9-s + (−0.258 + 0.965i)13-s + (−0.5 + 0.133i)19-s + (0.999 + i)21-s + (−0.707 + 0.707i)27-s + (−0.366 + 0.366i)31-s + (0.707 + 1.22i)37-s + (0.866 + 0.499i)39-s + (0.517 + 1.93i)43-s + (−0.499 − 0.866i)49-s + 0.517i·57-s + (0.866 − 1.5i)61-s + (1.22 − 0.707i)63-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (−0.707 + 1.22i)7-s + (−0.866 − 0.499i)9-s + (−0.258 + 0.965i)13-s + (−0.5 + 0.133i)19-s + (0.999 + i)21-s + (−0.707 + 0.707i)27-s + (−0.366 + 0.366i)31-s + (0.707 + 1.22i)37-s + (0.866 + 0.499i)39-s + (0.517 + 1.93i)43-s + (−0.499 − 0.866i)49-s + 0.517i·57-s + (0.866 − 1.5i)61-s + (1.22 − 0.707i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (2693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8975587904\)
\(L(\frac12)\) \(\approx\) \(0.8975587904\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
13 \( 1 + (0.258 - 0.965i)T \)
good7 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
37 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 - 1.93iT - T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
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

−8.645949978602020784708998832480, −8.182380670924121703580803494195, −7.21676703816070271648482465378, −6.51691217048852436005374468699, −6.07247862048133570806715847094, −5.21842876192888050778135345402, −4.13948118737622220824478475730, −2.99998972541840131545285896991, −2.44509486613916016686017245140, −1.45741491811639052297024400714, 0.47712682134215214163684716853, 2.27851964547833208871529416944, 3.26228808758928766518866203563, 3.88233768197271912471939827042, 4.56142442366950327719208655041, 5.50765467029635302799739839445, 6.18870767936792696205533737753, 7.27696909885039036002986480240, 7.68775297338897325027394675031, 8.689121323573631855989577272538

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