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Properties

Label 2-440-5.4-c1-0-4
Degree $2$
Conductor $440$
Sign $-0.382 - 0.924i$
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  + 2.67i·3-s + (2.06 − 0.854i)5-s + 4.38i·7-s − 4.16·9-s + 11-s − 4i·13-s + (2.28 + 5.53i)15-s + 5.87i·17-s + 0.526·19-s − 11.7·21-s − 6.15i·23-s + (3.53 − 3.53i)25-s − 3.11i·27-s + 0.967·29-s − 9.60·31-s + ⋯
L(s)  = 1  + 1.54i·3-s + (0.924 − 0.382i)5-s + 1.65i·7-s − 1.38·9-s + 0.301·11-s − 1.10i·13-s + (0.590 + 1.42i)15-s + 1.42i·17-s + 0.120·19-s − 2.56·21-s − 1.28i·23-s + (0.707 − 0.706i)25-s − 0.600i·27-s + 0.179·29-s − 1.72·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $-0.382 - 0.924i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ -0.382 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869503 + 1.30043i\)
\(L(\frac12)\) \(\approx\) \(0.869503 + 1.30043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.06 + 0.854i)T \)
11 \( 1 - T \)
good3 \( 1 - 2.67iT - 3T^{2} \)
7 \( 1 - 4.38iT - 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 5.87iT - 17T^{2} \)
19 \( 1 - 0.526T + 19T^{2} \)
23 \( 1 + 6.15iT - 23T^{2} \)
29 \( 1 - 0.967T + 29T^{2} \)
31 \( 1 + 9.60T + 31T^{2} \)
37 \( 1 - 1.76iT - 37T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 - 3.85iT - 43T^{2} \)
47 \( 1 + 4.91iT - 47T^{2} \)
53 \( 1 - 5.29iT - 53T^{2} \)
59 \( 1 - 2.63T + 59T^{2} \)
61 \( 1 - 8.96T + 61T^{2} \)
67 \( 1 + 7.97iT - 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 7.61iT - 83T^{2} \)
89 \( 1 + 2.83T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−11.09013232849132338965352625699, −10.37966067985155212483681827049, −9.596245338891264394229469523071, −8.876986011531523138153115959377, −8.306933722309562101498365159345, −6.21588864940382372254525555957, −5.57227261564228787467124010171, −4.82925200549877485150143948006, −3.47690279578739971815371405161, −2.23289473211796353211410446508, 1.07166490954719855280484516641, 2.16380909989744162391535422201, 3.72155750953396135761745437723, 5.29894867733515988941408702725, 6.56803429580549248334984396930, 7.09703745214647200148820793012, 7.60275048452195489756553018425, 9.115122928078444647443688869049, 9.877312332850492779370304626072, 11.07804146175942470602894751100

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