Properties

Label 2-440-11.5-c1-0-8
Degree $2$
Conductor $440$
Sign $-0.519 + 0.854i$
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  + (−0.582 − 0.422i)3-s + (−0.309 + 0.951i)5-s + (−3.38 + 2.45i)7-s + (−0.767 − 2.36i)9-s + (2.41 − 2.27i)11-s + (−1.86 − 5.75i)13-s + (0.582 − 0.422i)15-s + (2.01 − 6.20i)17-s + (−0.598 − 0.434i)19-s + 3.01·21-s − 5.13·23-s + (−0.809 − 0.587i)25-s + (−1.21 + 3.75i)27-s + (−1.68 + 1.22i)29-s + (−0.348 − 1.07i)31-s + ⋯
L(s)  = 1  + (−0.336 − 0.244i)3-s + (−0.138 + 0.425i)5-s + (−1.27 + 0.929i)7-s + (−0.255 − 0.786i)9-s + (0.727 − 0.686i)11-s + (−0.518 − 1.59i)13-s + (0.150 − 0.109i)15-s + (0.488 − 1.50i)17-s + (−0.137 − 0.0997i)19-s + 0.657·21-s − 1.06·23-s + (−0.161 − 0.117i)25-s + (−0.234 + 0.722i)27-s + (−0.312 + 0.227i)29-s + (−0.0625 − 0.192i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.519 + 0.854i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -0.519 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290667 - 0.516854i\)
\(L(\frac12)\) \(\approx\) \(0.290667 - 0.516854i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 + 2.27i)T \)
good3 \( 1 + (0.582 + 0.422i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (3.38 - 2.45i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.86 + 5.75i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.01 + 6.20i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.598 + 0.434i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.13T + 23T^{2} \)
29 \( 1 + (1.68 - 1.22i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.348 + 1.07i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.75 - 4.18i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.25 - 2.36i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.29T + 43T^{2} \)
47 \( 1 + (7.47 + 5.43i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.40 - 7.38i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.98 - 3.62i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.53 + 13.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 5.53T + 67T^{2} \)
71 \( 1 + (-2.74 + 8.43i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.54 + 4.75i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.176 + 0.542i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.70 - 14.4i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 1.33T + 89T^{2} \)
97 \( 1 + (-3.07 - 9.47i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.89088907333826060906145515270, −9.745445153200391340402748754858, −9.247488336852404832420634674938, −8.050966292851118222749609564902, −6.89242727655213229156639617584, −6.10544749798775029040937155070, −5.39903130654364963325250400796, −3.46968106017185697624032921088, −2.83881634176814498012872745373, −0.37512092212938964123103171068, 1.88515543515700679898598585022, 3.84660578885665919531452944724, 4.38493527000767104590226246831, 5.85214032634562243065085672295, 6.75099137763894441656380076378, 7.63857815896083414521801803772, 8.869704804817120237236948499077, 9.810395361812445619002113552637, 10.34307798104610607168126409495, 11.43822209571415422008894779518

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