↑

Properties

Label 2-3900-195.2-c0-0-1
Degree $2$
Conductor $3900$
Sign $0.998 + 0.0557i$
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  + (−0.965 − 0.258i)3-s + (1.67 + 0.965i)7-s + (0.866 + 0.499i)9-s + (−0.258 − 0.965i)13-s + (1.36 − 0.366i)19-s + (−1.36 − 1.36i)21-s + (−0.707 − 0.707i)27-s + (1.36 − 1.36i)31-s + (−1.22 + 0.707i)37-s + i·39-s + (−0.965 + 0.258i)43-s + (1.36 + 2.36i)49-s − 1.41·57-s + (−0.866 + 1.5i)61-s + (0.965 + 1.67i)63-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (1.67 + 0.965i)7-s + (0.866 + 0.499i)9-s + (−0.258 − 0.965i)13-s + (1.36 − 0.366i)19-s + (−1.36 − 1.36i)21-s + (−0.707 − 0.707i)27-s + (1.36 − 1.36i)31-s + (−1.22 + 0.707i)37-s + i·39-s + (−0.965 + 0.258i)43-s + (1.36 + 2.36i)49-s − 1.41·57-s + (−0.866 + 1.5i)61-s + (0.965 + 1.67i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.998 + 0.0557i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (1757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ 0.998 + 0.0557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210233460\)
\(L(\frac12)\) \(\approx\) \(1.210233460\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
13 \( 1 + (0.258 + 0.965i)T \)
good7 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
37 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 - 1.93T + T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.406725497769809946113083740694, −7.890253674391128792741571350673, −7.30505211485163484208594763686, −6.25996095654584722291066343629, −5.51982095423719457009382301530, −5.05819253378039406717328014106, −4.45996023989196285809138285379, −3.02647800321535592509334269489, −2.01622471854844382665386466981, −1.05604870789646734819954679946, 1.09897261282980558921428628998, 1.83245989793319091938964390291, 3.48530276289500404855344185016, 4.29554490493474011103365066572, 4.98529911777580686170068718575, 5.36061252058673806612168694948, 6.61752536636787412472882936470, 7.08661116432682762895976080508, 7.83213348316365448820601904930, 8.558147609769782897170112161796

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