↑

Properties

Label 2-1848-77.10-c1-0-32
Degree $2$
Conductor $1848$
Sign $0.999 - 0.0101i$
Analytic cond. $14.7563$
Root an. cond. $3.84140$
Motivic weight $1$
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)3-s + (1.74 − 1.00i)5-s + (2.55 − 0.701i)7-s + (0.499 + 0.866i)9-s + (−0.458 − 3.28i)11-s + 6.01·13-s + 2.01·15-s + (−3.93 + 6.81i)17-s + (3.66 + 6.34i)19-s + (2.55 + 0.668i)21-s + (−2.31 − 4.01i)23-s + (−0.471 + 0.816i)25-s + 0.999i·27-s + 2.60i·29-s + (5.08 + 2.93i)31-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (0.780 − 0.450i)5-s + (0.964 − 0.265i)7-s + (0.166 + 0.288i)9-s + (−0.138 − 0.990i)11-s + 1.66·13-s + 0.520·15-s + (−0.954 + 1.65i)17-s + (0.840 + 1.45i)19-s + (0.558 + 0.145i)21-s + (−0.483 − 0.836i)23-s + (−0.0943 + 0.163i)25-s + 0.192i·27-s + 0.483i·29-s + (0.913 + 0.527i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.999 - 0.0101i$
Analytic conductor: \(14.7563\)
Root analytic conductor: \(3.84140\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :1/2),\ 0.999 - 0.0101i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.872862108\)
\(L(\frac12)\) \(\approx\) \(2.872862108\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2.55 + 0.701i)T \)
11 \( 1 + (0.458 + 3.28i)T \)
good5 \( 1 + (-1.74 + 1.00i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 6.01T + 13T^{2} \)
17 \( 1 + (3.93 - 6.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.66 - 6.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.31 + 4.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.60iT - 29T^{2} \)
31 \( 1 + (-5.08 - 2.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.83 + 10.1i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.44T + 41T^{2} \)
43 \( 1 + 8.60iT - 43T^{2} \)
47 \( 1 + (-1.72 + 0.994i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.50 + 9.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.69 - 5.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.32 + 4.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.27 - 7.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.84T + 71T^{2} \)
73 \( 1 + (6.00 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.58 + 3.80i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.72T + 83T^{2} \)
89 \( 1 + (11.0 - 6.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.46iT - 97T^{2} \)
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

−8.857016725025193778801889326559, −8.563342374355315396223496032984, −8.105972851404011962422868651218, −6.80890245899291996862402495409, −5.80396366567785795378476090510, −5.40574591845830597263781126394, −4.02734392288985311160356482753, −3.61492196791158406244635505333, −2.00651607347857934921753657764, −1.30898061090220964856732758757, 1.27859690109978197663218309009, 2.27113630558751696701853822308, 3.02810225783488193790569085124, 4.42458346591963619562084752178, 5.10645171270937561638391528225, 6.19348679457503184253056702412, 6.89696024864146284277964728537, 7.66288671739666618715793309042, 8.506506595334079272359729263083, 9.240422048987683460253508844067

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