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Properties

Label 2-5-5.3-c20-0-6
Degree $2$
Conductor $5$
Sign $0.979 - 0.201i$
Analytic cond. $12.6756$
Root an. cond. $3.56029$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (836. − 836. i)2-s + (6.84e4 + 6.84e4i)3-s − 3.49e5i·4-s + (7.10e6 − 6.70e6i)5-s + 1.14e8·6-s + (−2.53e8 + 2.53e8i)7-s + (5.84e8 + 5.84e8i)8-s + 5.89e9i·9-s + (3.33e8 − 1.15e10i)10-s + 3.06e10·11-s + (2.39e10 − 2.39e10i)12-s + (−1.19e11 − 1.19e11i)13-s + 4.23e11i·14-s + (9.45e11 + 2.73e10i)15-s + 1.34e12·16-s + (4.41e11 − 4.41e11i)17-s + ⋯
L(s)  = 1  + (0.816 − 0.816i)2-s + (1.15 + 1.15i)3-s − 0.333i·4-s + (0.727 − 0.686i)5-s + 1.89·6-s + (−0.895 + 0.895i)7-s + (0.544 + 0.544i)8-s + 1.69i·9-s + (0.0333 − 1.15i)10-s + 1.18·11-s + (0.386 − 0.386i)12-s + (−0.864 − 0.864i)13-s + 1.46i·14-s + (1.63 + 0.0474i)15-s + 1.22·16-s + (0.219 − 0.219i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.979 - 0.201i$
Analytic conductor: \(12.6756\)
Root analytic conductor: \(3.56029\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :10),\ 0.979 - 0.201i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(4.13531 + 0.420963i\)
\(L(\frac12)\) \(\approx\) \(4.13531 + 0.420963i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.10e6 + 6.70e6i)T \)
good2 \( 1 + (-836. + 836. i)T - 1.04e6iT^{2} \)
3 \( 1 + (-6.84e4 - 6.84e4i)T + 3.48e9iT^{2} \)
7 \( 1 + (2.53e8 - 2.53e8i)T - 7.97e16iT^{2} \)
11 \( 1 - 3.06e10T + 6.72e20T^{2} \)
13 \( 1 + (1.19e11 + 1.19e11i)T + 1.90e22iT^{2} \)
17 \( 1 + (-4.41e11 + 4.41e11i)T - 4.06e24iT^{2} \)
19 \( 1 + 3.31e12iT - 3.75e25T^{2} \)
23 \( 1 + (-7.65e11 - 7.65e11i)T + 1.71e27iT^{2} \)
29 \( 1 + 1.13e14iT - 1.76e29T^{2} \)
31 \( 1 + 8.39e14T + 6.71e29T^{2} \)
37 \( 1 + (2.17e15 - 2.17e15i)T - 2.31e31iT^{2} \)
41 \( 1 + 7.50e15T + 1.80e32T^{2} \)
43 \( 1 + (2.71e16 + 2.71e16i)T + 4.67e32iT^{2} \)
47 \( 1 + (-3.40e16 + 3.40e16i)T - 2.76e33iT^{2} \)
53 \( 1 + (4.53e16 + 4.53e16i)T + 3.05e34iT^{2} \)
59 \( 1 - 7.34e17iT - 2.61e35T^{2} \)
61 \( 1 + 2.68e16T + 5.08e35T^{2} \)
67 \( 1 + (-1.71e18 + 1.71e18i)T - 3.32e36iT^{2} \)
71 \( 1 - 2.52e18T + 1.05e37T^{2} \)
73 \( 1 + (8.36e17 + 8.36e17i)T + 1.84e37iT^{2} \)
79 \( 1 + 3.07e18iT - 8.96e37T^{2} \)
83 \( 1 + (-6.40e18 - 6.40e18i)T + 2.40e38iT^{2} \)
89 \( 1 - 2.47e19iT - 9.72e38T^{2} \)
97 \( 1 + (9.21e19 - 9.21e19i)T - 5.43e39iT^{2} \)
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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

−19.65999398478725749370085189812, −16.79036658101434916613233758179, −15.09884861687873693801515080004, −13.70923721237110699403762848384, −12.31731175095417931628766585284, −9.950532701548081293395229888271, −8.837830146873198506748223326563, −5.11946808072265877972137283971, −3.51510451005447163187090023819, −2.31428979255738228090114988708, 1.59242723583644633540065588216, 3.56251750826001410579803099349, 6.49528187808159394661205131631, 7.18629548156289684763597842106, 9.659887479434744658849749233148, 12.85209683216211743174103107007, 14.05615686001015260049396387330, 14.53483404377581509554448018289, 16.86347014437139728704959646176, 18.91257287843092102485765943449

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