↑

Properties

Label 2-22e2-11.5-c1-0-6
Degree $2$
Conductor $484$
Sign $0.957 + 0.288i$
Analytic cond. $3.86475$
Root an. cond. $1.96589$
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  + (1.91 + 1.39i)3-s + (1.35 − 4.15i)5-s + (0.812 + 2.49i)9-s + (8.39 − 6.09i)15-s + 1.62·23-s + (−11.4 − 8.29i)25-s + (0.272 − 0.839i)27-s + (3.43 + 10.5i)31-s + (−4.13 + 3.00i)37-s + 11.4·45-s + (9.70 + 7.05i)47-s + (−2.16 + 6.65i)49-s + (1.85 + 5.70i)53-s + (−8.39 + 6.09i)59-s − 15.1·67-s + ⋯
L(s)  = 1  + (1.10 + 0.805i)3-s + (0.604 − 1.85i)5-s + (0.270 + 0.833i)9-s + (2.16 − 1.57i)15-s + 0.339·23-s + (−2.28 − 1.65i)25-s + (0.0525 − 0.161i)27-s + (0.616 + 1.89i)31-s + (−0.680 + 0.494i)37-s + 1.71·45-s + (1.41 + 1.02i)47-s + (−0.309 + 0.951i)49-s + (0.254 + 0.783i)53-s + (−1.09 + 0.793i)59-s − 1.84·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ 0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15407 - 0.316939i\)
\(L(\frac12)\) \(\approx\) \(2.15407 - 0.316939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-1.91 - 1.39i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-1.35 + 4.15i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.43 - 10.5i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.13 - 3.00i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-9.70 - 7.05i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.85 - 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.39 - 6.09i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + (-4.90 + 15.0i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 9.86T + 89T^{2} \)
97 \( 1 + (5.28 + 16.2i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.59001976626832087425828507358, −9.777620934439240700653246274641, −8.914351961883331936712112566205, −8.752207793632041366439639784155, −7.68667787055199871638499073181, −6.05033187097584387582550223055, −4.92271297021019258280638080830, −4.31299945899275917598747418302, −2.95063974319024041636477667266, −1.42377624086751629246938375523, 2.02717499625129186682031113407, 2.74802945733260367449269428671, 3.73238345763367543005521562789, 5.70414313086086085345462345777, 6.71486786494243242223797558064, 7.29508090061852291228894457677, 8.128110826820795577324372392624, 9.261301789486865173016784271716, 10.10513421755559296560263979036, 10.92768061784091117319877585238

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