↑

Properties

Label 2-464-116.99-c1-0-6
Degree $2$
Conductor $464$
Sign $0.672 - 0.739i$
Analytic cond. $3.70505$
Root an. cond. $1.92485$
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  + (1.30 − 1.30i)3-s + 3.62i·5-s + 1.31i·7-s − 0.406i·9-s + (−0.158 + 0.158i)11-s + 4.10i·13-s + (4.73 + 4.73i)15-s + (−1.97 − 1.97i)17-s + (0.759 − 0.759i)19-s + (1.72 + 1.72i)21-s + 3.15i·23-s − 8.13·25-s + (3.38 + 3.38i)27-s + (−0.639 − 5.34i)29-s + (6.58 − 6.58i)31-s + ⋯
L(s)  = 1  + (0.753 − 0.753i)3-s + 1.62i·5-s + 0.498i·7-s − 0.135i·9-s + (−0.0478 + 0.0478i)11-s + 1.13i·13-s + (1.22 + 1.22i)15-s + (−0.479 − 0.479i)17-s + (0.174 − 0.174i)19-s + (0.375 + 0.375i)21-s + 0.658i·23-s − 1.62·25-s + (0.651 + 0.651i)27-s + (−0.118 − 0.992i)29-s + (1.18 − 1.18i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.672 - 0.739i$
Analytic conductor: \(3.70505\)
Root analytic conductor: \(1.92485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1/2),\ 0.672 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57195 + 0.695221i\)
\(L(\frac12)\) \(\approx\) \(1.57195 + 0.695221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (0.639 + 5.34i)T \)
good3 \( 1 + (-1.30 + 1.30i)T - 3iT^{2} \)
5 \( 1 - 3.62iT - 5T^{2} \)
7 \( 1 - 1.31iT - 7T^{2} \)
11 \( 1 + (0.158 - 0.158i)T - 11iT^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 + (1.97 + 1.97i)T + 17iT^{2} \)
19 \( 1 + (-0.759 + 0.759i)T - 19iT^{2} \)
23 \( 1 - 3.15iT - 23T^{2} \)
31 \( 1 + (-6.58 + 6.58i)T - 31iT^{2} \)
37 \( 1 + (-0.338 + 0.338i)T - 37iT^{2} \)
41 \( 1 + (-0.459 + 0.459i)T - 41iT^{2} \)
43 \( 1 + (-7.46 + 7.46i)T - 43iT^{2} \)
47 \( 1 + (4.11 + 4.11i)T + 47iT^{2} \)
53 \( 1 - 7.59T + 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + (-3.51 - 3.51i)T + 61iT^{2} \)
67 \( 1 - 2.87T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 + (8.40 - 8.40i)T - 73iT^{2} \)
79 \( 1 + (8.03 - 8.03i)T - 79iT^{2} \)
83 \( 1 - 15.0iT - 83T^{2} \)
89 \( 1 + (7.68 + 7.68i)T + 89iT^{2} \)
97 \( 1 + (-3.15 + 3.15i)T - 97iT^{2} \)
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

−11.36770113396214843013146996959, −10.23878328291670734549092328227, −9.339869889250908775918466375284, −8.328985661609584039141049891103, −7.30390404255031212831566829916, −6.86466373433249160767371552828, −5.79023292749647554447685493735, −4.09840716860821675139893128836, −2.73806783289357424014306938547, −2.15843232149626265135839141059, 1.05365900026746372178944733465, 3.00808785884679229381246769654, 4.20702246067143134687363386681, 4.87182821043673754257108892945, 6.05213061868118060292366966498, 7.58878730106798418212474650062, 8.610123205406055368307273129566, 8.862435256468515798365169927673, 9.994826664867995722067383689536, 10.58811376381884648640130280528

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