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Properties

Label 2-20-20.19-c10-0-27
Degree $2$
Conductor $20$
Sign $-0.0758 - 0.997i$
Analytic cond. $12.7071$
Root an. cond. $3.56470$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32i·2-s − 1.02e3·4-s + (−237 − 3.11e3i)5-s + 3.27e4i·8-s − 5.90e4·9-s + (−9.97e4 + 7.58e3i)10-s + 2.91e5i·13-s + 1.04e6·16-s + 1.81e6i·17-s + 1.88e6i·18-s + (2.42e5 + 3.19e6i)20-s + (−9.65e6 + 1.47e6i)25-s + 9.32e6·26-s − 3.23e7·29-s − 3.35e7i·32-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.0758 − 0.997i)5-s + i·8-s − 0.999·9-s + (−0.997 + 0.0758i)10-s + 0.784i·13-s + 16-s + 1.27i·17-s + 0.999i·18-s + (0.0758 + 0.997i)20-s + (−0.988 + 0.151i)25-s + 0.784·26-s − 1.57·29-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0758 - 0.997i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0758 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.0758 - 0.997i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ -0.0758 - 0.997i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0163215 + 0.0176101i\)
\(L(\frac12)\) \(\approx\) \(0.0163215 + 0.0176101i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 32iT \)
5 \( 1 + (237 + 3.11e3i)T \)
good3 \( 1 + 5.90e4T^{2} \)
7 \( 1 + 2.82e8T^{2} \)
11 \( 1 - 2.59e10T^{2} \)
13 \( 1 - 2.91e5iT - 1.37e11T^{2} \)
17 \( 1 - 1.81e6iT - 2.01e12T^{2} \)
19 \( 1 - 6.13e12T^{2} \)
23 \( 1 + 4.14e13T^{2} \)
29 \( 1 + 3.23e7T + 4.20e14T^{2} \)
31 \( 1 - 8.19e14T^{2} \)
37 \( 1 + 1.38e8iT - 4.80e15T^{2} \)
41 \( 1 + 2.07e8T + 1.34e16T^{2} \)
43 \( 1 + 2.16e16T^{2} \)
47 \( 1 + 5.25e16T^{2} \)
53 \( 1 - 2.93e8iT - 1.74e17T^{2} \)
59 \( 1 - 5.11e17T^{2} \)
61 \( 1 - 1.33e9T + 7.13e17T^{2} \)
67 \( 1 + 1.82e18T^{2} \)
71 \( 1 - 3.25e18T^{2} \)
73 \( 1 + 1.78e9iT - 4.29e18T^{2} \)
79 \( 1 - 9.46e18T^{2} \)
83 \( 1 + 1.55e19T^{2} \)
89 \( 1 + 8.56e9T + 3.11e19T^{2} \)
97 \( 1 - 1.48e10iT - 7.37e19T^{2} \)
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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

−14.66048857301158082226361315217, −13.33683898830623415046039873339, −12.20674054119528787219085390849, −11.05668457554543724731985048082, −9.348816326607864177592522580533, −8.322223659141558793924353590275, −5.51166735838906271496744679882, −3.88620322910430489482607886180, −1.80879552975791149161111937351, −0.01010640148670726492398611891, 3.19718105483879006788069383338, 5.39170558775144095392667094485, 6.84170185753842172088754329020, 8.200229849834407582880479684846, 9.847991418686205980398288620904, 11.49265046338919852755875581141, 13.42479577316980359220625371892, 14.51370641964613354730091130475, 15.43458927908061322101577835386, 16.84809582052003103040373979644

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