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Properties

Label 2-5-5.4-c11-0-3
Degree $2$
Conductor $5$
Sign $-0.963 + 0.268i$
Analytic cond. $3.84171$
Root an. cond. $1.96002$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 58.2i·2-s − 258. i·3-s − 1.34e3·4-s + (−6.73e3 + 1.87e3i)5-s − 1.50e4·6-s − 2.16e4i·7-s − 4.07e4i·8-s + 1.10e5·9-s + (1.09e5 + 3.92e5i)10-s + 2.11e5·11-s + 3.49e5i·12-s − 2.27e6i·13-s − 1.26e6·14-s + (4.85e5 + 1.74e6i)15-s − 5.13e6·16-s + 4.99e6i·17-s + ⋯
L(s)  = 1  − 1.28i·2-s − 0.614i·3-s − 0.658·4-s + (−0.963 + 0.268i)5-s − 0.791·6-s − 0.487i·7-s − 0.439i·8-s + 0.622·9-s + (0.345 + 1.24i)10-s + 0.395·11-s + 0.405i·12-s − 1.69i·13-s − 0.628·14-s + (0.165 + 0.592i)15-s − 1.22·16-s + 0.852i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.963 + 0.268i$
Analytic conductor: \(3.84171\)
Root analytic conductor: \(1.96002\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :11/2),\ -0.963 + 0.268i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.172794 - 1.26320i\)
\(L(\frac12)\) \(\approx\) \(0.172794 - 1.26320i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (6.73e3 - 1.87e3i)T \)
good2 \( 1 + 58.2iT - 2.04e3T^{2} \)
3 \( 1 + 258. iT - 1.77e5T^{2} \)
7 \( 1 + 2.16e4iT - 1.97e9T^{2} \)
11 \( 1 - 2.11e5T + 2.85e11T^{2} \)
13 \( 1 + 2.27e6iT - 1.79e12T^{2} \)
17 \( 1 - 4.99e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.37e7T + 1.16e14T^{2} \)
23 \( 1 - 4.84e7iT - 9.52e14T^{2} \)
29 \( 1 - 5.94e7T + 1.22e16T^{2} \)
31 \( 1 + 7.04e7T + 2.54e16T^{2} \)
37 \( 1 + 4.76e7iT - 1.77e17T^{2} \)
41 \( 1 + 3.14e8T + 5.50e17T^{2} \)
43 \( 1 - 6.15e8iT - 9.29e17T^{2} \)
47 \( 1 + 2.06e9iT - 2.47e18T^{2} \)
53 \( 1 - 2.13e9iT - 9.26e18T^{2} \)
59 \( 1 + 5.96e8T + 3.01e19T^{2} \)
61 \( 1 - 6.12e9T + 4.35e19T^{2} \)
67 \( 1 - 1.60e9iT - 1.22e20T^{2} \)
71 \( 1 + 2.52e10T + 2.31e20T^{2} \)
73 \( 1 + 8.34e9iT - 3.13e20T^{2} \)
79 \( 1 + 8.16e9T + 7.47e20T^{2} \)
83 \( 1 - 1.64e10iT - 1.28e21T^{2} \)
89 \( 1 - 6.25e10T + 2.77e21T^{2} \)
97 \( 1 - 1.19e11iT - 7.15e21T^{2} \)
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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

−20.20838734101852690783516644197, −19.41951533377927567699170612133, −18.03977514948750897999297932343, −15.57648746291288845911898527219, −13.15797930572069634280622640906, −11.87208688395406558761158249453, −10.33065132480773488180558570911, −7.47133350031766548339475405272, −3.52181030592682174595392507615, −1.00661767972234499990234180571, 4.61707007540148267486996456753, 7.05477528110403963988064018014, 8.988237532802586584073545248335, 11.73261180275498378858120348516, 14.41290495317912526030809662431, 15.84081934322267907015587573956, 16.46658239052998940002446362547, 18.64045718392574622136536093266, 20.58023836038640206190214273956, 22.34625036521846348862232771699

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