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Properties

Label 2-5-5.4-c23-0-7
Degree $2$
Conductor $5$
Sign $-0.0713 + 0.997i$
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  + 3.62e3i·2-s − 4.21e5i·3-s − 4.75e6·4-s + (−7.79e6 + 1.08e8i)5-s + 1.52e9·6-s − 4.49e9i·7-s + 1.31e10i·8-s − 8.33e10·9-s + (−3.94e11 − 2.82e10i)10-s − 1.32e12·11-s + 2.00e12i·12-s − 8.78e12i·13-s + 1.63e13·14-s + (4.58e13 + 3.28e12i)15-s − 8.76e13·16-s − 1.40e14i·17-s + ⋯
L(s)  = 1  + 1.25i·2-s − 1.37i·3-s − 0.566·4-s + (−0.0713 + 0.997i)5-s + 1.71·6-s − 0.860i·7-s + 0.542i·8-s − 0.885·9-s + (−1.24 − 0.0893i)10-s − 1.40·11-s + 0.777i·12-s − 1.35i·13-s + 1.07·14-s + (1.36 + 0.0979i)15-s − 1.24·16-s − 0.996i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0713 + 0.997i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (-0.0713 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.0713 + 0.997i$
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: \((2,\ 5,\ (\ :23/2),\ -0.0713 + 0.997i)\)

Particular Values

\(L(12)\) \(\approx\) \(0.531783 - 0.571190i\)
\(L(\frac12)\) \(\approx\) \(0.531783 - 0.571190i\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (7.79e6 - 1.08e8i)T \)
good2 \( 1 - 3.62e3iT - 8.38e6T^{2} \)
3 \( 1 + 4.21e5iT - 9.41e10T^{2} \)
7 \( 1 + 4.49e9iT - 2.73e19T^{2} \)
11 \( 1 + 1.32e12T + 8.95e23T^{2} \)
13 \( 1 + 8.78e12iT - 4.17e25T^{2} \)
17 \( 1 + 1.40e14iT - 1.99e28T^{2} \)
19 \( 1 - 8.14e13T + 2.57e29T^{2} \)
23 \( 1 + 1.29e15iT - 2.08e31T^{2} \)
29 \( 1 + 2.61e16T + 4.31e33T^{2} \)
31 \( 1 + 2.15e17T + 2.00e34T^{2} \)
37 \( 1 + 1.82e18iT - 1.17e36T^{2} \)
41 \( 1 - 9.22e17T + 1.24e37T^{2} \)
43 \( 1 - 2.81e18iT - 3.71e37T^{2} \)
47 \( 1 - 1.85e19iT - 2.87e38T^{2} \)
53 \( 1 - 1.04e20iT - 4.55e39T^{2} \)
59 \( 1 + 9.90e19T + 5.36e40T^{2} \)
61 \( 1 - 4.58e20T + 1.15e41T^{2} \)
67 \( 1 - 5.55e20iT - 9.99e41T^{2} \)
71 \( 1 - 1.12e21T + 3.79e42T^{2} \)
73 \( 1 + 1.23e21iT - 7.18e42T^{2} \)
79 \( 1 + 1.14e22T + 4.42e43T^{2} \)
83 \( 1 + 4.00e21iT - 1.37e44T^{2} \)
89 \( 1 - 3.83e22T + 6.85e44T^{2} \)
97 \( 1 + 4.53e22iT - 4.96e45T^{2} \)
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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

−17.69348938653221294687197804135, −15.83597740267732546106684787126, −14.29948799850264793934893867743, −13.07264578435138791183577271525, −10.83021163936756147135375955759, −7.64765633853936850631700990259, −7.32285688108184980249867876466, −5.72582369240953057235792393676, −2.57898358975674294886551083000, −0.28027467530829087479155787865, 1.96370265505155018129326838630, 3.77075793258668919708552481626, 5.13943321787771404757541383030, 8.879374907540508107267350070235, 10.04394946588663141985352426566, 11.48902024671694672333275559062, 12.92874708341062062790486718917, 15.41098652396882438917520698424, 16.44706696621429671005523399194, 18.75823070136465076775998601893

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