↑

Properties

Label 2-462-33.2-c1-0-11
Degree $2$
Conductor $462$
Sign $0.706 + 0.707i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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  + (0.309 − 0.951i)2-s + (−1.51 + 0.834i)3-s + (−0.809 − 0.587i)4-s + (4.04 − 1.31i)5-s + (0.324 + 1.70i)6-s + (−0.587 + 0.809i)7-s + (−0.809 + 0.587i)8-s + (1.60 − 2.53i)9-s − 4.25i·10-s + (0.991 + 3.16i)11-s + (1.71 + 0.216i)12-s + (1.38 + 0.449i)13-s + (0.587 + 0.809i)14-s + (−5.04 + 5.37i)15-s + (0.309 + 0.951i)16-s + (−1.02 − 3.15i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.876 + 0.481i)3-s + (−0.404 − 0.293i)4-s + (1.80 − 0.587i)5-s + (0.132 + 0.694i)6-s + (−0.222 + 0.305i)7-s + (−0.286 + 0.207i)8-s + (0.535 − 0.844i)9-s − 1.34i·10-s + (0.298 + 0.954i)11-s + (0.496 + 0.0625i)12-s + (0.383 + 0.124i)13-s + (0.157 + 0.216i)14-s + (−1.30 + 1.38i)15-s + (0.0772 + 0.237i)16-s + (−0.248 − 0.765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.706 + 0.707i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.706 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43799 - 0.596380i\)
\(L(\frac12)\) \(\approx\) \(1.43799 - 0.596380i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (1.51 - 0.834i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (-0.991 - 3.16i)T \)
good5 \( 1 + (-4.04 + 1.31i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.38 - 0.449i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.02 + 3.15i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.26 - 1.73i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.63iT - 23T^{2} \)
29 \( 1 + (-7.71 - 5.60i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.07 + 6.39i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.11 + 1.53i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.55 - 2.58i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.85iT - 43T^{2} \)
47 \( 1 + (-3.47 - 4.78i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.422 - 0.137i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.66 - 6.41i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (14.2 - 4.63i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 0.736T + 67T^{2} \)
71 \( 1 + (8.22 - 2.67i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.33 - 1.84i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.55 - 0.829i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.71 - 8.35i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.590iT - 89T^{2} \)
97 \( 1 + (2.90 - 8.93i)T + (-78.4 - 57.0i)T^{2} \)
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

−10.67913740184239387658461655038, −10.15661596595651848424584363210, −9.402648898593121807019533710881, −8.840891497590851743670320599939, −6.76988135628065378996193651919, −6.00923845519654102798125759459, −5.13172365261447046744738375232, −4.39316960910286901329977868227, −2.59030503134987780543874479736, −1.28303206957145514900267225034, 1.44545387808459726783632864172, 3.11198375667099419162128019261, 4.86655618472468869116475359662, 5.93049192143229237366291001161, 6.25356615908818703438783630884, 7.05140861724428043712902569749, 8.337238400439271014990630936121, 9.464522729150593918272397588489, 10.33987008270362351040292198091, 11.00166109080551748008536500590

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