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Properties

Label 2-5-5.4-c9-0-1
Degree $2$
Conductor $5$
Sign $0.406 - 0.913i$
Analytic cond. $2.57517$
Root an. cond. $1.60473$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.843i·2-s + 179. i·3-s + 511.·4-s + (−568. + 1.27e3i)5-s − 151.·6-s − 8.71e3i·7-s + 863. i·8-s − 1.24e4·9-s + (−1.07e3 − 479. i)10-s + 4.45e4·11-s + 9.16e4i·12-s − 2.14e4i·13-s + 7.35e3·14-s + (−2.28e5 − 1.01e5i)15-s + 2.61e5·16-s − 3.00e5i·17-s + ⋯
L(s)  = 1  + 0.0372i·2-s + 1.27i·3-s + 0.998·4-s + (−0.406 + 0.913i)5-s − 0.0476·6-s − 1.37i·7-s + 0.0745i·8-s − 0.632·9-s + (−0.0340 − 0.0151i)10-s + 0.917·11-s + 1.27i·12-s − 0.208i·13-s + 0.0511·14-s + (−1.16 − 0.519i)15-s + 0.995·16-s − 0.871i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.406 - 0.913i$
Analytic conductor: \(2.57517\)
Root analytic conductor: \(1.60473\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :9/2),\ 0.406 - 0.913i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.25982 + 0.818238i\)
\(L(\frac12)\) \(\approx\) \(1.25982 + 0.818238i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (568. - 1.27e3i)T \)
good2 \( 1 - 0.843iT - 512T^{2} \)
3 \( 1 - 179. iT - 1.96e4T^{2} \)
7 \( 1 + 8.71e3iT - 4.03e7T^{2} \)
11 \( 1 - 4.45e4T + 2.35e9T^{2} \)
13 \( 1 + 2.14e4iT - 1.06e10T^{2} \)
17 \( 1 + 3.00e5iT - 1.18e11T^{2} \)
19 \( 1 + 5.65e5T + 3.22e11T^{2} \)
23 \( 1 - 9.50e5iT - 1.80e12T^{2} \)
29 \( 1 - 8.03e5T + 1.45e13T^{2} \)
31 \( 1 + 1.99e6T + 2.64e13T^{2} \)
37 \( 1 + 9.53e6iT - 1.29e14T^{2} \)
41 \( 1 + 2.54e7T + 3.27e14T^{2} \)
43 \( 1 + 2.32e7iT - 5.02e14T^{2} \)
47 \( 1 - 3.77e7iT - 1.11e15T^{2} \)
53 \( 1 + 4.79e7iT - 3.29e15T^{2} \)
59 \( 1 + 7.00e7T + 8.66e15T^{2} \)
61 \( 1 - 1.26e8T + 1.16e16T^{2} \)
67 \( 1 - 2.66e8iT - 2.72e16T^{2} \)
71 \( 1 - 6.59e7T + 4.58e16T^{2} \)
73 \( 1 - 1.47e7iT - 5.88e16T^{2} \)
79 \( 1 - 4.66e7T + 1.19e17T^{2} \)
83 \( 1 - 2.01e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.54e8T + 3.50e17T^{2} \)
97 \( 1 + 3.39e8iT - 7.60e17T^{2} \)
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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

−22.13157700713439453435668565862, −20.70070861482831749089040171645, −19.59015321128470958522070642103, −16.98643246344756846455524648869, −15.73086688659783430906865798029, −14.46261027238274674268282166665, −11.27038605898789196311788413999, −10.21196647938936155377652079279, −7.05083715819696954166285473446, −3.73921493109274457842441622083, 1.76511031061647646631543012965, 6.37816424446454197007858084399, 8.413826415017186276113723017656, 11.84098952926041106032905428276, 12.64232969756892670754402226057, 15.17597332297266355341544155098, 16.88656601889865950722610501348, 18.83088772989177603272164560100, 19.83773674695818149447442954709, 21.52816635080805168348419066953

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