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Properties

Label 2-5-5.4-c23-0-9
Degree $2$
Conductor $5$
Sign $0.650 - 0.759i$
Analytic cond. $16.7602$
Root an. cond. $4.09392$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.56e3i·2-s − 4.21e5i·3-s − 2.25e7·4-s + (7.09e7 − 8.29e7i)5-s − 2.34e9·6-s + 7.17e8i·7-s + 7.86e10i·8-s − 8.38e10·9-s + (−4.61e11 − 3.94e11i)10-s − 2.81e11·11-s + 9.50e12i·12-s − 8.58e12i·13-s + 3.99e12·14-s + (−3.49e13 − 2.99e13i)15-s + 2.48e14·16-s − 7.16e13i·17-s + ⋯
L(s)  = 1  − 1.92i·2-s − 1.37i·3-s − 2.68·4-s + (0.650 − 0.759i)5-s − 2.63·6-s + 0.137i·7-s + 3.23i·8-s − 0.890·9-s + (−1.45 − 1.24i)10-s − 0.297·11-s + 3.69i·12-s − 1.32i·13-s + 0.263·14-s + (−1.04 − 0.894i)15-s + 3.53·16-s − 0.506i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.650 - 0.759i$
Analytic conductor: \(16.7602\)
Root analytic conductor: \(4.09392\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :23/2),\ 0.650 - 0.759i)\)

Particular Values

\(L(12)\) \(\approx\) \(1.20185 + 0.553313i\)
\(L(\frac12)\) \(\approx\) \(1.20185 + 0.553313i\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.09e7 + 8.29e7i)T \)
good2 \( 1 + 5.56e3iT - 8.38e6T^{2} \)
3 \( 1 + 4.21e5iT - 9.41e10T^{2} \)
7 \( 1 - 7.17e8iT - 2.73e19T^{2} \)
11 \( 1 + 2.81e11T + 8.95e23T^{2} \)
13 \( 1 + 8.58e12iT - 4.17e25T^{2} \)
17 \( 1 + 7.16e13iT - 1.99e28T^{2} \)
19 \( 1 - 3.73e14T + 2.57e29T^{2} \)
23 \( 1 - 2.05e15iT - 2.08e31T^{2} \)
29 \( 1 - 5.28e16T + 4.31e33T^{2} \)
31 \( 1 + 6.19e16T + 2.00e34T^{2} \)
37 \( 1 - 1.32e18iT - 1.17e36T^{2} \)
41 \( 1 + 5.55e18T + 1.24e37T^{2} \)
43 \( 1 - 4.87e18iT - 3.71e37T^{2} \)
47 \( 1 - 8.62e18iT - 2.87e38T^{2} \)
53 \( 1 + 7.45e19iT - 4.55e39T^{2} \)
59 \( 1 - 3.38e20T + 5.36e40T^{2} \)
61 \( 1 - 8.12e19T + 1.15e41T^{2} \)
67 \( 1 + 2.97e20iT - 9.99e41T^{2} \)
71 \( 1 + 1.10e21T + 3.79e42T^{2} \)
73 \( 1 + 9.17e20iT - 7.18e42T^{2} \)
79 \( 1 - 3.90e21T + 4.42e43T^{2} \)
83 \( 1 + 2.44e21iT - 1.37e44T^{2} \)
89 \( 1 + 4.42e22T + 6.85e44T^{2} \)
97 \( 1 - 3.47e22iT - 4.96e45T^{2} \)
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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

−17.62706097277261188748972698487, −13.66799469471476175643220848837, −12.92091435400650154197645244373, −11.87558695131896711046274318191, −9.968427746717523290381329965749, −8.287230024089981402259672895227, −5.25713199583928173903618689496, −2.81851011341904562378685316293, −1.49241645011175374261357116489, −0.55947827765812649645455555179, 3.97465694762847944970936948394, 5.40626756172563702937420344557, 6.92246526481113955313839795785, 8.989625889664466499554691989929, 10.19270381815815990600097258566, 13.83767491449029715371028916411, 14.83697232655852260185350655336, 16.00907894460793846571921794598, 17.06381016103924560702375917577, 18.51842284349126081396774570337

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