↑

Properties

Label 2-440-11.9-c1-0-7
Degree $2$
Conductor $440$
Sign $0.0138 + 0.999i$
Analytic cond. $3.51341$
Root an. cond. $1.87441$
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  + (−1.52 + 1.10i)3-s + (0.309 + 0.951i)5-s + (−2.97 − 2.16i)7-s + (0.166 − 0.511i)9-s + (−0.923 − 3.18i)11-s + (0.721 − 2.22i)13-s + (−1.52 − 1.10i)15-s + (−0.0990 − 0.304i)17-s + (3.66 − 2.66i)19-s + 6.91·21-s − 0.780·23-s + (−0.809 + 0.587i)25-s + (−1.43 − 4.40i)27-s + (−1.23 − 0.894i)29-s + (1.54 − 4.74i)31-s + ⋯
L(s)  = 1  + (−0.878 + 0.638i)3-s + (0.138 + 0.425i)5-s + (−1.12 − 0.817i)7-s + (0.0553 − 0.170i)9-s + (−0.278 − 0.960i)11-s + (0.200 − 0.615i)13-s + (−0.392 − 0.285i)15-s + (−0.0240 − 0.0739i)17-s + (0.840 − 0.610i)19-s + 1.50·21-s − 0.162·23-s + (−0.161 + 0.117i)25-s + (−0.275 − 0.847i)27-s + (−0.228 − 0.166i)29-s + (0.276 − 0.851i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0138 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0138 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.0138 + 0.999i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ 0.0138 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367298 - 0.362243i\)
\(L(\frac12)\) \(\approx\) \(0.367298 - 0.362243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.923 + 3.18i)T \)
good3 \( 1 + (1.52 - 1.10i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (2.97 + 2.16i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.721 + 2.22i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.0990 + 0.304i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.66 + 2.66i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.780T + 23T^{2} \)
29 \( 1 + (1.23 + 0.894i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.54 + 4.74i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.16 + 5.20i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.86 + 3.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.236T + 43T^{2} \)
47 \( 1 + (10.8 - 7.87i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.83 + 8.73i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.89 + 3.55i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.566 + 1.74i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.56T + 67T^{2} \)
71 \( 1 + (-4.00 - 12.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.56 + 4.77i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.47 - 10.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.34 - 13.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.85T + 89T^{2} \)
97 \( 1 + (1.72 - 5.29i)T + (-78.4 - 57.0i)T^{2} \)
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

−10.91669586087516151399632527122, −10.12106158724805213064230493960, −9.486137929043009394971953873362, −8.091472537310691515315890616143, −7.00728381066155175370276254254, −6.05118828270702462486682082540, −5.30323180605359140562038656846, −3.93635662422519058820261910520, −2.95912922188507247504429871620, −0.35988407640562081069616683947, 1.63197582396578458511627416570, 3.24745148195074866486454922169, 4.84896867358839007198398687013, 5.83933677438016164778427817798, 6.53470699409950101236926850179, 7.42594573333488481060045387401, 8.781915635352426967831398228123, 9.552671189178186576173007600995, 10.39484997677261872046718819973, 11.79196635367456724690095660289

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