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Properties

Label 2-440-440.259-c1-0-43
Degree $2$
Conductor $440$
Sign $0.999 - 0.0141i$
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  + (1.34 − 0.437i)2-s + (1.61 − 1.17i)4-s + (−0.690 + 2.12i)5-s + (−0.200 − 0.275i)7-s + (1.66 − 2.28i)8-s + (0.927 + 2.85i)9-s + 3.16i·10-s + (2.66 + 1.97i)11-s + (2.38 − 0.774i)13-s + (−0.389 − 0.282i)14-s + (1.23 − 3.80i)16-s + (2.49 + 3.43i)18-s + (4.50 − 6.19i)19-s + (1.38 + 4.25i)20-s + (4.44 + 1.48i)22-s − 8.60·23-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.0756 − 0.104i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)9-s + 1.00i·10-s + (0.804 + 0.594i)11-s + (0.660 − 0.214i)13-s + (−0.104 − 0.0756i)14-s + (0.309 − 0.951i)16-s + (0.587 + 0.809i)18-s + (1.03 − 1.42i)19-s + (0.309 + 0.951i)20-s + (0.948 + 0.316i)22-s − 1.79·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.999 - 0.0141i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ 0.999 - 0.0141i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47538 + 0.0174564i\)
\(L(\frac12)\) \(\approx\) \(2.47538 + 0.0174564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.437i)T \)
5 \( 1 + (0.690 - 2.12i)T \)
11 \( 1 + (-2.66 - 1.97i)T \)
good3 \( 1 + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (0.200 + 0.275i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-2.38 + 0.774i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.50 + 6.19i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 8.60T + 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.58 - 5.51i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.70 - 9.23i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-1.17 - 0.855i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.32 + 7.16i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.36 + 6.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 18.7T + 89T^{2} \)
97 \( 1 + (78.4 - 57.0i)T^{2} \)
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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

−11.43782328900261860033160874139, −10.33819333893623470294642512594, −9.813590182826767481759817007150, −8.149195092354990103344696449972, −7.10252641023449417811001259145, −6.50960492011154187384678965832, −5.23129837159166865913821498465, −4.16692135682555610063785593071, −3.16398282622042239773129041865, −1.87027916377681365579235092414, 1.51597078879713109635375841180, 3.66917865278753865332377848570, 3.98814517482070216346668134100, 5.53933042854246160265794278924, 6.13254713946289709326070993605, 7.31048060376434282961380230465, 8.336791764691874785196124565601, 9.115455128489647318584456573807, 10.29924557793221166161887515234, 11.69727794051683298557495949231

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