↑

Properties

Label 2-3675-21.2-c0-0-13
Degree $2$
Conductor $3675$
Sign $0.982 - 0.188i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
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  + (0.866 + 0.5i)2-s + (0.965 + 0.258i)3-s + (0.707 + 0.707i)6-s − i·8-s + (0.866 + 0.499i)9-s + (0.866 − 0.5i)11-s − 1.41·13-s + (0.5 − 0.866i)16-s + (1.22 − 0.707i)17-s + (0.5 + 0.866i)18-s + 0.999·22-s + (−0.866 − 0.5i)23-s + (0.258 − 0.965i)24-s + (−1.22 − 0.707i)26-s + (0.707 + 0.707i)27-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.965 + 0.258i)3-s + (0.707 + 0.707i)6-s − i·8-s + (0.866 + 0.499i)9-s + (0.866 − 0.5i)11-s − 1.41·13-s + (0.5 − 0.866i)16-s + (1.22 − 0.707i)17-s + (0.5 + 0.866i)18-s + 0.999·22-s + (−0.866 − 0.5i)23-s + (0.258 − 0.965i)24-s + (−1.22 − 0.707i)26-s + (0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.982 - 0.188i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.982 - 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.763316750\)
\(L(\frac12)\) \(\approx\) \(2.763316750\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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

−8.841747307753403674700813658776, −7.67109513451859361052745850212, −7.39427261279431096771284435236, −6.43819387734568530972489122379, −5.65633355891408140465085672839, −4.80570003028843598866454580829, −4.24904555359280955996702526816, −3.37954727658765970461432211318, −2.64696323198864025968511512521, −1.25473568343187657848978236910, 1.62430878052678876931228914385, 2.37971691273080649180534669148, 3.24510002881566207448555937634, 3.99129635932580318885005263597, 4.53964302991665438914838641201, 5.53442987796266569850630379080, 6.41739270996413063209104450442, 7.49211375817195530229080123992, 7.81698931607121278524544504271, 8.612135005639810800720168099740

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