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Properties

Label 2-20-20.3-c5-0-2
Degree $2$
Conductor $20$
Sign $0.778 - 0.627i$
Analytic cond. $3.20767$
Root an. cond. $1.79099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.91 − 4.08i)2-s + (−6.89 + 6.89i)3-s + (−1.39 + 31.9i)4-s + (53.4 + 16.2i)5-s + (55.1 + 1.20i)6-s + (57.8 + 57.8i)7-s + (136. − 119. i)8-s + 147. i·9-s + (−142. − 282. i)10-s + 628. i·11-s + (−210. − 230. i)12-s + (−402. − 402. i)13-s + (10.0 − 462. i)14-s + (−481. + 256. i)15-s + (−1.02e3 − 89.1i)16-s + (366. − 366. i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.722i)2-s + (−0.442 + 0.442i)3-s + (−0.0435 + 0.999i)4-s + (0.956 + 0.291i)5-s + (0.625 + 0.0136i)6-s + (0.446 + 0.446i)7-s + (0.751 − 0.659i)8-s + 0.608i·9-s + (−0.451 − 0.892i)10-s + 1.56i·11-s + (−0.422 − 0.461i)12-s + (−0.661 − 0.661i)13-s + (0.0137 − 0.630i)14-s + (−0.552 + 0.294i)15-s + (−0.996 − 0.0870i)16-s + (0.307 − 0.307i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(3.20767\)
Root analytic conductor: \(1.79099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5/2),\ 0.778 - 0.627i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.916208 + 0.323075i\)
\(L(\frac12)\) \(\approx\) \(0.916208 + 0.323075i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.91 + 4.08i)T \)
5 \( 1 + (-53.4 - 16.2i)T \)
good3 \( 1 + (6.89 - 6.89i)T - 243iT^{2} \)
7 \( 1 + (-57.8 - 57.8i)T + 1.68e4iT^{2} \)
11 \( 1 - 628. iT - 1.61e5T^{2} \)
13 \( 1 + (402. + 402. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-366. + 366. i)T - 1.41e6iT^{2} \)
19 \( 1 - 1.24e3T + 2.47e6T^{2} \)
23 \( 1 + (-12.5 + 12.5i)T - 6.43e6iT^{2} \)
29 \( 1 - 3.63e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.71e3iT - 2.86e7T^{2} \)
37 \( 1 + (-6.66e3 + 6.66e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 9.44e3T + 1.15e8T^{2} \)
43 \( 1 + (7.76e3 - 7.76e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.95e3 + 1.95e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (-1.79e4 - 1.79e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 4.57e4T + 7.14e8T^{2} \)
61 \( 1 - 2.50e4T + 8.44e8T^{2} \)
67 \( 1 + (3.51e4 + 3.51e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 4.40e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.35e4 + 2.35e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 4.32e4T + 3.07e9T^{2} \)
83 \( 1 + (-2.37e4 + 2.37e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.96e4iT - 5.58e9T^{2} \)
97 \( 1 + (-92.6 + 92.6i)T - 8.58e9iT^{2} \)
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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

−17.63906239555272111035538859167, −16.59904720480571655971773403230, −14.92934350367502729385282005954, −13.17384281258960434674336094774, −11.79060901194245104874295133394, −10.36049532858427162898387082432, −9.557809575817677976996719132537, −7.53460378602330097665141777768, −5.05992692409594022490606466249, −2.20841621936683709902757149705, 1.06009784924904773549270961873, 5.49561551789530556130287543459, 6.77360540942228643985899777533, 8.585702473635544936101582226617, 9.996183251003358657426574704246, 11.56506286392173230766824921362, 13.53852231967168297900004294406, 14.50906118754562103789142311379, 16.35450707117383830647661496541, 17.18847078111345025473327643163

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