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Properties

Label 2-440-11.5-c1-0-6
Degree $2$
Conductor $440$
Sign $0.752 - 0.659i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.35 + 1.70i)3-s + (0.309 − 0.951i)5-s + (1.76 − 1.28i)7-s + (1.68 + 5.17i)9-s + (−2.98 + 1.44i)11-s + (1.90 + 5.85i)13-s + (2.35 − 1.70i)15-s + (2.20 − 6.77i)17-s + (−5.14 − 3.73i)19-s + 6.34·21-s + 2.17·23-s + (−0.809 − 0.587i)25-s + (−2.18 + 6.73i)27-s + (2.58 − 1.87i)29-s + (1.35 + 4.17i)31-s + ⋯
L(s)  = 1  + (1.35 + 0.985i)3-s + (0.138 − 0.425i)5-s + (0.667 − 0.484i)7-s + (0.560 + 1.72i)9-s + (−0.899 + 0.436i)11-s + (0.528 + 1.62i)13-s + (0.606 − 0.440i)15-s + (0.534 − 1.64i)17-s + (−1.18 − 0.857i)19-s + 1.38·21-s + 0.454·23-s + (−0.161 − 0.117i)25-s + (−0.421 + 1.29i)27-s + (0.479 − 0.348i)29-s + (0.243 + 0.750i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.752 - 0.659i$
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: \((2,\ 440,\ (\ :1/2),\ 0.752 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09130 + 0.786790i\)
\(L(\frac12)\) \(\approx\) \(2.09130 + 0.786790i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.98 - 1.44i)T \)
good3 \( 1 + (-2.35 - 1.70i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.76 + 1.28i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.90 - 5.85i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.20 + 6.77i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.14 + 3.73i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.17T + 23T^{2} \)
29 \( 1 + (-2.58 + 1.87i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.35 - 4.17i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.28 - 3.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.65 + 1.93i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.83T + 43T^{2} \)
47 \( 1 + (-3.48 - 2.53i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.28 + 10.0i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (11.1 - 8.10i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.27 + 6.99i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.67T + 67T^{2} \)
71 \( 1 + (-1.77 + 5.46i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.96 + 2.87i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.50 + 7.71i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.30 + 4.01i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (1.01 + 3.11i)T + (-78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.98753560005289613922763847793, −10.16656539990760214752259534956, −9.269585691958663578192164380660, −8.722890381199783657452540446555, −7.82384050995307461460440431170, −6.81625974303236631816584148435, −4.75720963865138197203255206750, −4.64957378586283048196049284994, −3.19022751094921992980104008319, −1.97801967935142203907742920592, 1.60784780823344360041870802958, 2.73212306210564655950369259556, 3.64572402786103204267350483494, 5.52204823801292160749067788949, 6.40263933320785890223641908921, 7.82299290458962681407088219232, 8.132684239183452680828961660035, 8.746196434706842038769528219487, 10.26436152589052770173059930751, 10.78561282110593464341349004632

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