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Properties

Label 2-5-5.4-c21-0-2
Degree $2$
Conductor $5$
Sign $-0.471 - 0.881i$
Analytic cond. $13.9738$
Root an. cond. $3.73816$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 883. i·2-s + 1.98e5i·3-s + 1.31e6·4-s + (1.03e7 + 1.92e7i)5-s + 1.75e8·6-s + 4.00e8i·7-s − 3.01e9i·8-s − 2.91e10·9-s + (1.70e10 − 9.10e9i)10-s + 7.41e10·11-s + 2.61e11i·12-s + 1.00e11i·13-s + 3.53e11·14-s + (−3.83e12 + 2.05e12i)15-s + 9.39e10·16-s + 1.87e12i·17-s + ⋯
L(s)  = 1  − 0.610i·2-s + 1.94i·3-s + 0.627·4-s + (0.471 + 0.881i)5-s + 1.18·6-s + 0.535i·7-s − 0.993i·8-s − 2.78·9-s + (0.538 − 0.287i)10-s + 0.861·11-s + 1.22i·12-s + 0.202i·13-s + 0.327·14-s + (−1.71 + 0.918i)15-s + 0.0213·16-s + 0.226i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.471 - 0.881i$
Analytic conductor: \(13.9738\)
Root analytic conductor: \(3.73816\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :21/2),\ -0.471 - 0.881i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.08755 + 1.81551i\)
\(L(\frac12)\) \(\approx\) \(1.08755 + 1.81551i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.03e7 - 1.92e7i)T \)
good2 \( 1 + 883. iT - 2.09e6T^{2} \)
3 \( 1 - 1.98e5iT - 1.04e10T^{2} \)
7 \( 1 - 4.00e8iT - 5.58e17T^{2} \)
11 \( 1 - 7.41e10T + 7.40e21T^{2} \)
13 \( 1 - 1.00e11iT - 2.47e23T^{2} \)
17 \( 1 - 1.87e12iT - 6.90e25T^{2} \)
19 \( 1 + 1.90e13T + 7.14e26T^{2} \)
23 \( 1 - 1.08e13iT - 3.94e28T^{2} \)
29 \( 1 + 2.62e15T + 5.13e30T^{2} \)
31 \( 1 - 5.54e15T + 2.08e31T^{2} \)
37 \( 1 - 1.29e15iT - 8.55e32T^{2} \)
41 \( 1 - 1.05e17T + 7.38e33T^{2} \)
43 \( 1 + 4.65e16iT - 2.00e34T^{2} \)
47 \( 1 - 3.00e17iT - 1.30e35T^{2} \)
53 \( 1 - 1.17e18iT - 1.62e36T^{2} \)
59 \( 1 - 5.18e18T + 1.54e37T^{2} \)
61 \( 1 - 3.02e18T + 3.10e37T^{2} \)
67 \( 1 + 4.51e18iT - 2.22e38T^{2} \)
71 \( 1 - 7.47e18T + 7.52e38T^{2} \)
73 \( 1 + 3.42e19iT - 1.34e39T^{2} \)
79 \( 1 + 1.06e20T + 7.08e39T^{2} \)
83 \( 1 - 1.09e20iT - 1.99e40T^{2} \)
89 \( 1 - 1.63e20T + 8.65e40T^{2} \)
97 \( 1 - 5.30e20iT - 5.27e41T^{2} \)
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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

−19.36218377971295354262070012476, −17.02903207424711796864176566066, −15.56501919768809852479205899575, −14.57308619929312457404305422856, −11.54392613955282747675445342723, −10.50371546022679429261110489204, −9.326879729836111320169015156579, −6.05915098610761985229948954743, −3.87575714633550712988058944967, −2.53076853390985652758433974733, 0.874692455228041254225000445496, 2.11042779637063926664906819903, 5.88401223773956871259071389727, 7.05476680255647683021458725948, 8.416899616516625931316879058703, 11.65322882161317784433706619507, 12.99361047686343707643626535093, 14.30911035744189145733786380123, 16.78545647261216035705192389980, 17.57381504468963585243929245339

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