Properties

Label 2-20-4.3-c10-0-18
Degree $2$
Conductor $20$
Sign $-0.805 - 0.593i$
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  + (9.98 − 30.4i)2-s − 330. i·3-s + (−824. − 607. i)4-s + 1.39e3·5-s + (−1.00e4 − 3.30e3i)6-s − 2.94e4i·7-s + (−2.66e4 + 1.89e4i)8-s − 5.03e4·9-s + (1.39e4 − 4.24e4i)10-s + 1.22e5i·11-s + (−2.00e5 + 2.72e5i)12-s + 4.49e5·13-s + (−8.95e5 − 2.94e5i)14-s − 4.62e5i·15-s + (3.10e5 + 1.00e6i)16-s − 1.32e5·17-s + ⋯
L(s)  = 1  + (0.312 − 0.950i)2-s − 1.36i·3-s + (−0.805 − 0.593i)4-s + 0.447·5-s + (−1.29 − 0.424i)6-s − 1.75i·7-s + (−0.814 + 0.579i)8-s − 0.852·9-s + (0.139 − 0.424i)10-s + 0.760i·11-s + (−0.807 + 1.09i)12-s + 1.21·13-s + (−1.66 − 0.546i)14-s − 0.608i·15-s + (0.296 + 0.955i)16-s − 0.0930·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.805 - 0.593i$
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: \((2,\ 20,\ (\ :5),\ -0.805 - 0.593i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.580360 + 1.76636i\)
\(L(\frac12)\) \(\approx\) \(0.580360 + 1.76636i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.98 + 30.4i)T \)
5 \( 1 - 1.39e3T \)
good3 \( 1 + 330. iT - 5.90e4T^{2} \)
7 \( 1 + 2.94e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.22e5iT - 2.59e10T^{2} \)
13 \( 1 - 4.49e5T + 1.37e11T^{2} \)
17 \( 1 + 1.32e5T + 2.01e12T^{2} \)
19 \( 1 - 3.03e6iT - 6.13e12T^{2} \)
23 \( 1 + 4.11e6iT - 4.14e13T^{2} \)
29 \( 1 + 2.05e7T + 4.20e14T^{2} \)
31 \( 1 + 4.65e7iT - 8.19e14T^{2} \)
37 \( 1 - 7.43e7T + 4.80e15T^{2} \)
41 \( 1 - 1.29e7T + 1.34e16T^{2} \)
43 \( 1 - 4.13e7iT - 2.16e16T^{2} \)
47 \( 1 + 1.40e8iT - 5.25e16T^{2} \)
53 \( 1 + 7.63e8T + 1.74e17T^{2} \)
59 \( 1 - 3.73e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.40e9T + 7.13e17T^{2} \)
67 \( 1 + 1.39e9iT - 1.82e18T^{2} \)
71 \( 1 - 7.53e8iT - 3.25e18T^{2} \)
73 \( 1 - 1.69e8T + 4.29e18T^{2} \)
79 \( 1 - 3.47e9iT - 9.46e18T^{2} \)
83 \( 1 - 8.27e8iT - 1.55e19T^{2} \)
89 \( 1 - 6.94e9T + 3.11e19T^{2} \)
97 \( 1 - 1.99e8T + 7.37e19T^{2} \)
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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

−14.40107087215751008969071594942, −13.41452377878906663843833710872, −12.80300820043072958611157408676, −11.19796715098632071371611734731, −9.930840822722839445120892709584, −7.83632731931861287922559854589, −6.28787381711504527085117076235, −4.00131956971245034316925712481, −1.82323770016742638615210692211, −0.77166829588807780910886678613, 3.25645707460155799953597005292, 5.06267408115672415204571492890, 6.06723716788325481230180027907, 8.676051850066849166354518791835, 9.335875841262103810776845143951, 11.27536714646484188427539196920, 13.07836923331613126339951382258, 14.56194852903854449076133180434, 15.66999432947101942191630253062, 16.07701710150984934441983068320

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