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Properties

Label 2-440-11.3-c1-0-5
Degree $2$
Conductor $440$
Sign $0.820 - 0.570i$
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  + (0.529 + 1.62i)3-s + (0.809 + 0.587i)5-s + (1.14 − 3.52i)7-s + (0.0536 − 0.0389i)9-s + (2.68 + 1.94i)11-s + (0.952 − 0.692i)13-s + (−0.529 + 1.62i)15-s + (4.36 + 3.17i)17-s + (−1.18 − 3.65i)19-s + 6.34·21-s − 8.68·23-s + (0.309 + 0.951i)25-s + (4.24 + 3.08i)27-s + (−2.12 + 6.53i)29-s + (−7.08 + 5.14i)31-s + ⋯
L(s)  = 1  + (0.305 + 0.940i)3-s + (0.361 + 0.262i)5-s + (0.432 − 1.33i)7-s + (0.0178 − 0.0129i)9-s + (0.810 + 0.585i)11-s + (0.264 − 0.191i)13-s + (−0.136 + 0.420i)15-s + (1.05 + 0.769i)17-s + (−0.272 − 0.839i)19-s + 1.38·21-s − 1.81·23-s + (0.0618 + 0.190i)25-s + (0.817 + 0.594i)27-s + (−0.394 + 1.21i)29-s + (−1.27 + 0.924i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.820 - 0.570i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ 0.820 - 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71492 + 0.537695i\)
\(L(\frac12)\) \(\approx\) \(1.71492 + 0.537695i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-2.68 - 1.94i)T \)
good3 \( 1 + (-0.529 - 1.62i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-1.14 + 3.52i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.952 + 0.692i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.36 - 3.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.18 + 3.65i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.68T + 23T^{2} \)
29 \( 1 + (2.12 - 6.53i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.08 - 5.14i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.696 + 2.14i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.493 - 1.51i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.11T + 43T^{2} \)
47 \( 1 + (3.91 + 12.0i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.0 + 7.27i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.121 - 0.374i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.45 + 1.05i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + (-5.54 - 4.02i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.10 + 9.56i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.901 + 0.654i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.0140 + 0.0101i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.49T + 89T^{2} \)
97 \( 1 + (8.50 - 6.18i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.77802096324504929252496811314, −10.35959349950188664317805983155, −9.608495191798892867399336398659, −8.656023377561716388975600557299, −7.47529630496870875874596484630, −6.67350207509499830348837136317, −5.29829743066175301441536608567, −4.09273527567767794195186501598, −3.59867020135047573339833956219, −1.58158974515677073654275041805, 1.50493210030109633307638404028, 2.48263114074135971625730581968, 4.10574102380384410045250750766, 5.71374935846870344837063606204, 6.09907006578978306870078695924, 7.57538660540593717241792506215, 8.204465633104089888598574085722, 9.122956538219368981369589004392, 9.928460863612931037479943490545, 11.35309119394465541419047217236

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