↑

Properties

Label 2-468-117.110-c1-0-7
Degree $2$
Conductor $468$
Sign $0.996 + 0.0865i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (−0.867 + 1.49i)3-s + (0.308 − 1.15i)5-s + (−1.69 − 1.69i)7-s + (−1.49 − 2.60i)9-s + (5.49 + 1.47i)11-s + (3.57 − 0.462i)13-s + (1.46 + 1.46i)15-s + (−2.93 − 5.08i)17-s + (2.50 + 0.671i)19-s + (4.00 − 1.06i)21-s + 1.88·23-s + (3.09 + 1.78i)25-s + (5.19 + 0.0159i)27-s + (−5.12 + 2.95i)29-s + (5.65 + 1.51i)31-s + ⋯
L(s)  = 1  + (−0.500 + 0.865i)3-s + (0.138 − 0.515i)5-s + (−0.639 − 0.639i)7-s + (−0.498 − 0.867i)9-s + (1.65 + 0.443i)11-s + (0.991 − 0.128i)13-s + (0.376 + 0.377i)15-s + (−0.711 − 1.23i)17-s + (0.574 + 0.153i)19-s + (0.874 − 0.233i)21-s + 0.393·23-s + (0.619 + 0.357i)25-s + (0.999 + 0.00307i)27-s + (−0.951 + 0.549i)29-s + (1.01 + 0.272i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.996 + 0.0865i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.996 + 0.0865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23953 - 0.0537435i\)
\(L(\frac12)\) \(\approx\) \(1.23953 - 0.0537435i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.867 - 1.49i)T \)
13 \( 1 + (-3.57 + 0.462i)T \)
good5 \( 1 + (-0.308 + 1.15i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.69 + 1.69i)T + 7iT^{2} \)
11 \( 1 + (-5.49 - 1.47i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.93 + 5.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.50 - 0.671i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 1.88T + 23T^{2} \)
29 \( 1 + (5.12 - 2.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.65 - 1.51i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.18 + 1.65i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.51 + 3.51i)T + 41iT^{2} \)
43 \( 1 - 2.37iT - 43T^{2} \)
47 \( 1 + (-0.781 - 2.91i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 9.59iT - 53T^{2} \)
59 \( 1 + (1.78 + 6.66i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + (4.32 - 4.32i)T - 67iT^{2} \)
71 \( 1 + (-1.56 + 5.82i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (11.8 + 11.8i)T + 73iT^{2} \)
79 \( 1 + (5.60 - 9.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.3 - 3.02i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-3.51 - 13.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.23 - 6.23i)T - 97iT^{2} \)
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

−11.13976437190676276000815283012, −9.967834965902229713525256205492, −9.337035068179125610193665210373, −8.727167841410685256666746858028, −7.04420018007133971397661545305, −6.39173142108957914079809695022, −5.18763993330094981528535659845, −4.21011318224575172497342056536, −3.35020337418720249540517352397, −1.00992181883288380086561238116, 1.34841602210571373405086793260, 2.83805886954639543178264148128, 4.15432410648112190995867273748, 5.96783205459795360787201278790, 6.22715795245637301731889448954, 7.07998221381417547450682633100, 8.440664913289305696200966552222, 9.065426315821102644618259832634, 10.29508394347434130633434972784, 11.35935551777173363463134306019

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