Properties

Label 2-5-5.4-c11-0-1
Degree $2$
Conductor $5$
Sign $0.941 - 0.336i$
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  − 27.1i·2-s + 524. i·3-s + 1.31e3·4-s + (6.58e3 − 2.34e3i)5-s + 1.42e4·6-s + 6.65e4i·7-s − 9.11e4i·8-s − 9.74e4·9-s + (−6.36e4 − 1.78e5i)10-s − 3.74e5·11-s + 6.88e5i·12-s − 1.12e6i·13-s + 1.80e6·14-s + (1.23e6 + 3.44e6i)15-s + 2.19e5·16-s − 5.82e6i·17-s + ⋯
L(s)  = 1  − 0.598i·2-s + 1.24i·3-s + 0.641·4-s + (0.941 − 0.336i)5-s + 0.745·6-s + 1.49i·7-s − 0.983i·8-s − 0.550·9-s + (−0.201 − 0.564i)10-s − 0.701·11-s + 0.798i·12-s − 0.842i·13-s + 0.896·14-s + (0.418 + 1.17i)15-s + 0.0523·16-s − 0.995i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.941 - 0.336i$
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: \((2,\ 5,\ (\ :11/2),\ 0.941 - 0.336i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.80350 + 0.312238i\)
\(L(\frac12)\) \(\approx\) \(1.80350 + 0.312238i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-6.58e3 + 2.34e3i)T \)
good2 \( 1 + 27.1iT - 2.04e3T^{2} \)
3 \( 1 - 524. iT - 1.77e5T^{2} \)
7 \( 1 - 6.65e4iT - 1.97e9T^{2} \)
11 \( 1 + 3.74e5T + 2.85e11T^{2} \)
13 \( 1 + 1.12e6iT - 1.79e12T^{2} \)
17 \( 1 + 5.82e6iT - 3.42e13T^{2} \)
19 \( 1 + 6.03e6T + 1.16e14T^{2} \)
23 \( 1 + 7.51e6iT - 9.52e14T^{2} \)
29 \( 1 - 4.87e7T + 1.22e16T^{2} \)
31 \( 1 + 2.71e8T + 2.54e16T^{2} \)
37 \( 1 - 5.42e5iT - 1.77e17T^{2} \)
41 \( 1 - 8.20e8T + 5.50e17T^{2} \)
43 \( 1 + 6.03e8iT - 9.29e17T^{2} \)
47 \( 1 + 4.69e8iT - 2.47e18T^{2} \)
53 \( 1 - 3.82e9iT - 9.26e18T^{2} \)
59 \( 1 - 2.19e9T + 3.01e19T^{2} \)
61 \( 1 + 7.28e9T + 4.35e19T^{2} \)
67 \( 1 - 2.11e9iT - 1.22e20T^{2} \)
71 \( 1 + 4.86e9T + 2.31e20T^{2} \)
73 \( 1 - 1.92e10iT - 3.13e20T^{2} \)
79 \( 1 - 4.54e10T + 7.47e20T^{2} \)
83 \( 1 - 3.61e10iT - 1.28e21T^{2} \)
89 \( 1 + 3.43e9T + 2.77e21T^{2} \)
97 \( 1 - 1.57e11iT - 7.15e21T^{2} \)
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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

−21.34013141592300633957127443427, −20.44087886639177031148948284775, −18.35403744138338378299683213272, −16.22511796141559962757842273609, −15.18211435239499784176360148444, −12.58328044898187592834337900047, −10.65656439512370713046430257390, −9.301902220300958234914018109563, −5.47498220054405900199036787634, −2.58077226329377286389339677664, 1.78066607609686079653940534680, 6.40059444104325970438097964571, 7.52752175144386911386732019154, 10.72882256818281727153010647193, 13.07661773660912250004618234644, 14.34680017363561147144175830881, 16.68577318064180969201896407240, 17.82461611815596898866144476115, 19.52306703035445631300284236245, 21.11414033069353010734234912877

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