Properties

Label 2-20-4.3-c10-0-10
Degree $2$
Conductor $20$
Sign $0.998 + 0.0481i$
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  + (−0.770 + 31.9i)2-s − 80.5i·3-s + (−1.02e3 − 49.2i)4-s − 1.39e3·5-s + (2.57e3 + 62.0i)6-s + 345. i·7-s + (2.36e3 − 3.26e4i)8-s + 5.25e4·9-s + (1.07e3 − 4.47e4i)10-s − 1.67e5i·11-s + (−3.97e3 + 8.23e4i)12-s + 9.56e4·13-s + (−1.10e4 − 265. i)14-s + 1.12e5i·15-s + (1.04e6 + 1.00e5i)16-s + 2.26e6·17-s + ⋯
L(s)  = 1  + (−0.0240 + 0.999i)2-s − 0.331i·3-s + (−0.998 − 0.0481i)4-s − 0.447·5-s + (0.331 + 0.00798i)6-s + 0.0205i·7-s + (0.0721 − 0.997i)8-s + 0.890·9-s + (0.0107 − 0.447i)10-s − 1.03i·11-s + (−0.0159 + 0.331i)12-s + 0.257·13-s + (−0.0205 − 0.000494i)14-s + 0.148i·15-s + (0.995 + 0.0961i)16-s + 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0481i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.998 + 0.0481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.998 + 0.0481i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ 0.998 + 0.0481i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.45245 - 0.0349727i\)
\(L(\frac12)\) \(\approx\) \(1.45245 - 0.0349727i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.770 - 31.9i)T \)
5 \( 1 + 1.39e3T \)
good3 \( 1 + 80.5iT - 5.90e4T^{2} \)
7 \( 1 - 345. iT - 2.82e8T^{2} \)
11 \( 1 + 1.67e5iT - 2.59e10T^{2} \)
13 \( 1 - 9.56e4T + 1.37e11T^{2} \)
17 \( 1 - 2.26e6T + 2.01e12T^{2} \)
19 \( 1 - 1.35e6iT - 6.13e12T^{2} \)
23 \( 1 + 7.72e6iT - 4.14e13T^{2} \)
29 \( 1 + 6.87e6T + 4.20e14T^{2} \)
31 \( 1 + 4.09e7iT - 8.19e14T^{2} \)
37 \( 1 + 4.60e7T + 4.80e15T^{2} \)
41 \( 1 - 1.22e7T + 1.34e16T^{2} \)
43 \( 1 + 2.14e8iT - 2.16e16T^{2} \)
47 \( 1 + 1.55e8iT - 5.25e16T^{2} \)
53 \( 1 + 5.32e7T + 1.74e17T^{2} \)
59 \( 1 - 9.05e8iT - 5.11e17T^{2} \)
61 \( 1 - 7.30e8T + 7.13e17T^{2} \)
67 \( 1 - 5.67e8iT - 1.82e18T^{2} \)
71 \( 1 - 2.07e9iT - 3.25e18T^{2} \)
73 \( 1 + 3.04e9T + 4.29e18T^{2} \)
79 \( 1 + 2.49e9iT - 9.46e18T^{2} \)
83 \( 1 - 6.00e9iT - 1.55e19T^{2} \)
89 \( 1 - 5.14e9T + 3.11e19T^{2} \)
97 \( 1 - 6.01e9T + 7.37e19T^{2} \)
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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

−16.03125106535093661540883357247, −14.73982784217501584284426717290, −13.50464281483564882792239087210, −12.22899060481132593434738501896, −10.19936557868201066563637333573, −8.475913240268328952424141066154, −7.31139199027536008267158121896, −5.77271898414712113302052769978, −3.87213034790779576871240809266, −0.75942169178873004930748584442, 1.38065740275025408233515069864, 3.49506769006339495257475154397, 4.90905720587079563938890041507, 7.57935372751909775167771635966, 9.404431411588235023846322873102, 10.44950195484161635173096932957, 11.92462519939699400932152478546, 12.96193139855379796796679622227, 14.53349446345603714764185702236, 15.87565635545164845263854360906

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