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Properties

Label 2-5-5.3-c20-0-2
Degree $2$
Conductor $5$
Sign $-0.991 - 0.130i$
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  + (3.77 − 3.77i)2-s + (3.88e4 + 3.88e4i)3-s + 1.04e6i·4-s + (−8.90e6 + 4.00e6i)5-s + 2.93e5·6-s + (4.46e7 − 4.46e7i)7-s + (7.91e6 + 7.91e6i)8-s − 4.66e8i·9-s + (−1.84e7 + 4.87e7i)10-s − 1.60e10·11-s + (−4.07e10 + 4.07e10i)12-s + (−3.76e10 − 3.76e10i)13-s − 3.37e8i·14-s + (−5.01e11 − 1.90e11i)15-s − 1.09e12·16-s + (−2.08e12 + 2.08e12i)17-s + ⋯
L(s)  = 1  + (0.00368 − 0.00368i)2-s + (0.658 + 0.658i)3-s + 0.999i·4-s + (−0.911 + 0.410i)5-s + 0.00484·6-s + (0.158 − 0.158i)7-s + (0.00736 + 0.00736i)8-s − 0.133i·9-s + (−0.00184 + 0.00487i)10-s − 0.620·11-s + (−0.658 + 0.658i)12-s + (−0.273 − 0.273i)13-s − 0.00116i·14-s + (−0.870 − 0.329i)15-s − 0.999·16-s + (−1.03 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.991 - 0.130i$
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: \((2,\ 5,\ (\ :10),\ -0.991 - 0.130i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.0730680 + 1.11819i\)
\(L(\frac12)\) \(\approx\) \(0.0730680 + 1.11819i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (8.90e6 - 4.00e6i)T \)
good2 \( 1 + (-3.77 + 3.77i)T - 1.04e6iT^{2} \)
3 \( 1 + (-3.88e4 - 3.88e4i)T + 3.48e9iT^{2} \)
7 \( 1 + (-4.46e7 + 4.46e7i)T - 7.97e16iT^{2} \)
11 \( 1 + 1.60e10T + 6.72e20T^{2} \)
13 \( 1 + (3.76e10 + 3.76e10i)T + 1.90e22iT^{2} \)
17 \( 1 + (2.08e12 - 2.08e12i)T - 4.06e24iT^{2} \)
19 \( 1 - 2.99e11iT - 3.75e25T^{2} \)
23 \( 1 + (-4.11e13 - 4.11e13i)T + 1.71e27iT^{2} \)
29 \( 1 - 3.65e14iT - 1.76e29T^{2} \)
31 \( 1 + 1.33e15T + 6.71e29T^{2} \)
37 \( 1 + (1.48e15 - 1.48e15i)T - 2.31e31iT^{2} \)
41 \( 1 - 2.38e16T + 1.80e32T^{2} \)
43 \( 1 + (-5.02e15 - 5.02e15i)T + 4.67e32iT^{2} \)
47 \( 1 + (1.96e16 - 1.96e16i)T - 2.76e33iT^{2} \)
53 \( 1 + (6.48e15 + 6.48e15i)T + 3.05e34iT^{2} \)
59 \( 1 - 5.84e17iT - 2.61e35T^{2} \)
61 \( 1 - 9.66e17T + 5.08e35T^{2} \)
67 \( 1 + (2.23e18 - 2.23e18i)T - 3.32e36iT^{2} \)
71 \( 1 - 1.33e18T + 1.05e37T^{2} \)
73 \( 1 + (4.76e18 + 4.76e18i)T + 1.84e37iT^{2} \)
79 \( 1 - 1.15e19iT - 8.96e37T^{2} \)
83 \( 1 + (6.54e18 + 6.54e18i)T + 2.40e38iT^{2} \)
89 \( 1 - 2.48e19iT - 9.72e38T^{2} \)
97 \( 1 + (-4.02e19 + 4.02e19i)T - 5.43e39iT^{2} \)
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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

−19.70659833661465611377627409775, −17.77188790149747466307266626171, −15.98088153676943994200935442216, −14.84932260179677593000488716110, −12.78625751276252719979940975667, −11.00000639960031500262202382299, −8.807218228606728549757801127441, −7.41047931274465667264543783133, −4.15243536479070323781632468272, −2.98994724394567330638011177770, 0.43026022227495722232061114622, 2.26066441428551692386591223093, 4.87814619421104410102048972674, 7.25154492095403180351288526850, 8.903197692978131090179589868455, 11.10934213340723085041926597021, 13.07353312445326029563685469335, 14.59306155501068849535034646710, 15.99752612327750236774296393703, 18.44829097555261198740027829557

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