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Properties

Label 2-20-20.3-c19-0-27
Degree $2$
Conductor $20$
Sign $0.0837 - 0.996i$
Analytic cond. $45.7633$
Root an. cond. $6.76486$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−408. + 597. i)2-s + (1.30e4 − 1.30e4i)3-s + (−1.89e5 − 4.88e5i)4-s + (4.34e6 − 4.33e5i)5-s + (2.45e6 + 1.31e7i)6-s + (1.13e8 + 1.13e8i)7-s + (3.69e8 + 8.64e7i)8-s + 8.23e8i·9-s + (−1.51e9 + 2.77e9i)10-s − 3.35e9i·11-s + (−8.83e9 − 3.89e9i)12-s + (2.30e9 + 2.30e9i)13-s + (−1.14e11 + 2.14e10i)14-s + (5.09e10 − 6.22e10i)15-s + (−2.02e11 + 1.85e11i)16-s + (−7.82e10 + 7.82e10i)17-s + ⋯
L(s)  = 1  + (−0.564 + 0.825i)2-s + (0.381 − 0.381i)3-s + (−0.361 − 0.932i)4-s + (0.995 − 0.0991i)5-s + (0.0994 + 0.530i)6-s + (1.06 + 1.06i)7-s + (0.973 + 0.227i)8-s + 0.708i·9-s + (−0.480 + 0.877i)10-s − 0.428i·11-s + (−0.494 − 0.217i)12-s + (0.0603 + 0.0603i)13-s + (−1.48 + 0.277i)14-s + (0.342 − 0.417i)15-s + (−0.737 + 0.674i)16-s + (−0.160 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0837 - 0.996i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.0837 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.0837 - 0.996i$
Analytic conductor: \(45.7633\)
Root analytic conductor: \(6.76486\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :19/2),\ 0.0837 - 0.996i)\)

Particular Values

\(L(10)\) \(\approx\) \(2.379449143\)
\(L(\frac12)\) \(\approx\) \(2.379449143\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (408. - 597. i)T \)
5 \( 1 + (-4.34e6 + 4.33e5i)T \)
good3 \( 1 + (-1.30e4 + 1.30e4i)T - 1.16e9iT^{2} \)
7 \( 1 + (-1.13e8 - 1.13e8i)T + 1.13e16iT^{2} \)
11 \( 1 + 3.35e9iT - 6.11e19T^{2} \)
13 \( 1 + (-2.30e9 - 2.30e9i)T + 1.46e21iT^{2} \)
17 \( 1 + (7.82e10 - 7.82e10i)T - 2.39e23iT^{2} \)
19 \( 1 - 1.81e12T + 1.97e24T^{2} \)
23 \( 1 + (8.76e12 - 8.76e12i)T - 7.46e25iT^{2} \)
29 \( 1 + 5.98e13iT - 6.10e27T^{2} \)
31 \( 1 + 1.34e14iT - 2.16e28T^{2} \)
37 \( 1 + (2.75e14 - 2.75e14i)T - 6.24e29iT^{2} \)
41 \( 1 - 2.32e14T + 4.39e30T^{2} \)
43 \( 1 + (3.48e15 - 3.48e15i)T - 1.08e31iT^{2} \)
47 \( 1 + (2.69e15 + 2.69e15i)T + 5.88e31iT^{2} \)
53 \( 1 + (-2.77e16 - 2.77e16i)T + 5.77e32iT^{2} \)
59 \( 1 - 2.50e16T + 4.42e33T^{2} \)
61 \( 1 + 2.30e16T + 8.34e33T^{2} \)
67 \( 1 + (1.60e17 + 1.60e17i)T + 4.95e34iT^{2} \)
71 \( 1 - 2.11e17iT - 1.49e35T^{2} \)
73 \( 1 + (-5.60e17 - 5.60e17i)T + 2.53e35iT^{2} \)
79 \( 1 - 1.21e18T + 1.13e36T^{2} \)
83 \( 1 + (1.72e18 - 1.72e18i)T - 2.90e36iT^{2} \)
89 \( 1 + 2.50e18iT - 1.09e37T^{2} \)
97 \( 1 + (-6.37e18 + 6.37e18i)T - 5.60e37iT^{2} \)
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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

−14.18810678144237887017101898995, −13.47403687319981564643999831552, −11.44173725376362129376506593210, −9.839954551105124742470575025791, −8.632514102695186804205206717085, −7.70054742105654131319446046839, −5.93324520028074615175603631392, −5.07727917069887874441761593039, −2.24744653486663785671516473665, −1.35153862068362089432006762534, 0.817723556293573787425734713435, 1.91701415808858237411935149839, 3.45224371533054907634633559643, 4.80215195918010492877174928929, 7.08230829428780519883180534460, 8.579446696450088036061464564808, 9.814276910378098992962217791265, 10.63159320113208287051706624900, 12.09760850869161000184244245880, 13.66095576872428435693725865638

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