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Properties

Label 2-5e2-5.4-c23-0-13
Degree $2$
Conductor $25$
Sign $-0.447 + 0.894i$
Analytic cond. $83.8010$
Root an. cond. $9.15428$
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  − 4.01e3i·2-s − 3.88e5i·3-s − 7.74e6·4-s − 1.56e9·6-s + 3.81e9i·7-s − 2.59e9i·8-s − 5.67e10·9-s + 2.52e11·11-s + 3.00e12i·12-s + 3.59e12i·13-s + 1.53e13·14-s − 7.53e13·16-s + 2.34e14i·17-s + 2.27e14i·18-s + 6.23e14·19-s + ⋯
L(s)  = 1  − 1.38i·2-s − 1.26i·3-s − 0.922·4-s − 1.75·6-s + 0.728i·7-s − 0.106i·8-s − 0.602·9-s + 0.266·11-s + 1.16i·12-s + 0.555i·13-s + 1.01·14-s − 1.07·16-s + 1.65i·17-s + 0.835i·18-s + 1.22·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(83.8010\)
Root analytic conductor: \(9.15428\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :23/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(12)\) \(\approx\) \(2.228220093\)
\(L(\frac12)\) \(\approx\) \(2.228220093\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + 4.01e3iT - 8.38e6T^{2} \)
3 \( 1 + 3.88e5iT - 9.41e10T^{2} \)
7 \( 1 - 3.81e9iT - 2.73e19T^{2} \)
11 \( 1 - 2.52e11T + 8.95e23T^{2} \)
13 \( 1 - 3.59e12iT - 4.17e25T^{2} \)
17 \( 1 - 2.34e14iT - 1.99e28T^{2} \)
19 \( 1 - 6.23e14T + 2.57e29T^{2} \)
23 \( 1 - 3.58e15iT - 2.08e31T^{2} \)
29 \( 1 - 2.05e16T + 4.31e33T^{2} \)
31 \( 1 - 1.36e17T + 2.00e34T^{2} \)
37 \( 1 + 1.23e18iT - 1.17e36T^{2} \)
41 \( 1 - 1.40e18T + 1.24e37T^{2} \)
43 \( 1 + 2.18e17iT - 3.71e37T^{2} \)
47 \( 1 + 8.67e18iT - 2.87e38T^{2} \)
53 \( 1 - 7.63e19iT - 4.55e39T^{2} \)
59 \( 1 - 1.01e18T + 5.36e40T^{2} \)
61 \( 1 - 2.87e20T + 1.15e41T^{2} \)
67 \( 1 - 1.47e21iT - 9.99e41T^{2} \)
71 \( 1 - 7.64e20T + 3.79e42T^{2} \)
73 \( 1 - 3.49e21iT - 7.18e42T^{2} \)
79 \( 1 + 1.02e22T + 4.42e43T^{2} \)
83 \( 1 - 7.71e21iT - 1.37e44T^{2} \)
89 \( 1 + 4.58e21T + 6.85e44T^{2} \)
97 \( 1 + 1.13e23iT - 4.96e45T^{2} \)
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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

−12.17643823852844431776805895455, −11.40705842153588220354556313745, −9.940361610065515413597591997791, −8.642169660205956911778377855090, −7.17647951106763633777640768805, −5.90007001688794699306719860778, −3.97550960170846634562508585791, −2.61407428783408381156921792797, −1.68706114807931210531232549502, −0.943440395284269295984630745978, 0.64434854550797597207831616198, 3.03741612663149515479373649001, 4.49664052789471709141799739962, 5.23815604006697873882148537014, 6.72373770817790202572748477947, 7.81769286734576439169860443900, 9.202416722090054290308590717521, 10.21659700795131982446134186626, 11.56129758773940425773924468469, 13.64418620759739133381501339365

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