↑

Properties

Label 2-252-63.59-c1-0-2
Degree $2$
Conductor $252$
Sign $0.333 - 0.942i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  + (−1.44 + 0.956i)3-s + 2.86·5-s + (−1.83 + 1.90i)7-s + (1.16 − 2.76i)9-s + 2.71i·11-s + (3.18 + 1.84i)13-s + (−4.14 + 2.74i)15-s + (−3.22 + 5.58i)17-s + (2.73 − 1.58i)19-s + (0.819 − 4.50i)21-s + 2.99i·23-s + 3.22·25-s + (0.956 + 5.10i)27-s + (−2.48 + 1.43i)29-s + (8.26 − 4.77i)31-s + ⋯
L(s)  = 1  + (−0.833 + 0.552i)3-s + 1.28·5-s + (−0.692 + 0.721i)7-s + (0.389 − 0.920i)9-s + 0.817i·11-s + (0.884 + 0.510i)13-s + (−1.06 + 0.708i)15-s + (−0.781 + 1.35i)17-s + (0.628 − 0.362i)19-s + (0.178 − 0.983i)21-s + 0.623i·23-s + 0.645·25-s + (0.184 + 0.982i)27-s + (−0.461 + 0.266i)29-s + (1.48 − 0.857i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.333 - 0.942i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.333 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895735 + 0.633002i\)
\(L(\frac12)\) \(\approx\) \(0.895735 + 0.633002i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 - 0.956i)T \)
7 \( 1 + (1.83 - 1.90i)T \)
good5 \( 1 - 2.86T + 5T^{2} \)
11 \( 1 - 2.71iT - 11T^{2} \)
13 \( 1 + (-3.18 - 1.84i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.73 + 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.99iT - 23T^{2} \)
29 \( 1 + (2.48 - 1.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.26 + 4.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.70 + 2.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.794 + 1.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.67 + 8.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.65 + 9.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.16 + 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.33 + 7.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.566 + 0.327i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.86 + 6.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 + (-11.0 - 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.59 - 4.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.92 - 13.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.14 + 5.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.2 + 7.62i)T + (48.5 - 84.0i)T^{2} \)
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

−12.22098052788197778068104064145, −11.20979091063230654891913959505, −10.17613314178145382939984657440, −9.558605874804241169145186151600, −8.750611949308542118242388861734, −6.78094676122083384320858498481, −6.08920872187191469956700076458, −5.26698114846211107486518284173, −3.83403524570532299934229589893, −1.97378115836210442940779006582, 1.05252846935472732246611356843, 2.91342194540165627773136029488, 4.81249755793926844035795945020, 6.04993006030177818650852659555, 6.47695228548924264582142928113, 7.73741561579912983269925195646, 9.123166896265861921226830650373, 10.14616503341330169342081196421, 10.82147713189595846595764862145, 11.82695141609553522328974245247

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