↑

Properties

Label 2-1848-1848.1187-c0-0-4
Degree $2$
Conductor $1848$
Sign $0.444 + 0.895i$
Analytic cond. $0.922272$
Root an. cond. $0.960350$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 1.5i)5-s − 0.999·6-s + (0.866 − 0.5i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.5i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + (−0.866 − 0.499i)14-s + 1.73·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 1.5i)5-s − 0.999·6-s + (0.866 − 0.5i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.5i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + (−0.866 − 0.499i)14-s + 1.73·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(0.922272\)
Root analytic conductor: \(0.960350\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :0),\ 0.444 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286640010\)
\(L(\frac12)\) \(\approx\) \(1.286640010\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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

−9.349972656055222346811887419307, −8.525115930788390202759903333843, −7.83561056494335896442814478261, −7.01610556720469995436537437179, −6.45982951435982865675863135463, −5.32283808579801896318915417201, −3.71019649105072726450865136061, −3.21897578913793341031607602839, −2.09881579263313082159341949513, −1.44680004969862980030673143438, 1.40774965471252286284790932503, 2.41094726674489170760290760670, 4.47352504971145478490373938595, 4.70469376399237820858168167714, 5.34500653823643821908356193681, 6.29160735062614467400458895453, 7.42079156349673222564371880835, 8.387981945105118661022295282628, 8.815263872239862809340897909759, 9.298471094629011961961419218985

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