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Properties

Label 2-5e2-5.4-c19-0-24
Degree $2$
Conductor $25$
Sign $-0.447 + 0.894i$
Analytic cond. $57.2041$
Root an. cond. $7.56334$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 456i·2-s − 5.06e4i·3-s + 3.16e5·4-s + 2.30e7·6-s − 1.69e7i·7-s + 3.83e8i·8-s − 1.40e9·9-s − 1.62e7·11-s − 1.60e10i·12-s − 5.04e10i·13-s + 7.71e9·14-s − 8.93e9·16-s + 2.25e11i·17-s − 6.39e11i·18-s + 1.71e12·19-s + ⋯
L(s)  = 1  + 0.629i·2-s − 1.48i·3-s + 0.603·4-s + 0.935·6-s − 0.158i·7-s + 1.00i·8-s − 1.20·9-s − 0.00207·11-s − 0.896i·12-s − 1.31i·13-s + 0.0997·14-s − 0.0325·16-s + 0.460i·17-s − 0.760i·18-s + 1.21·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(57.2041\)
Root analytic conductor: \(7.56334\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :19/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(10)\) \(\approx\) \(2.073306133\)
\(L(\frac12)\) \(\approx\) \(2.073306133\)
\(L(\frac{21}{2})\) 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 \( 1 - 456iT - 5.24e5T^{2} \)
3 \( 1 + 5.06e4iT - 1.16e9T^{2} \)
7 \( 1 + 1.69e7iT - 1.13e16T^{2} \)
11 \( 1 + 1.62e7T + 6.11e19T^{2} \)
13 \( 1 + 5.04e10iT - 1.46e21T^{2} \)
17 \( 1 - 2.25e11iT - 2.39e23T^{2} \)
19 \( 1 - 1.71e12T + 1.97e24T^{2} \)
23 \( 1 + 1.40e13iT - 7.46e25T^{2} \)
29 \( 1 + 1.13e12T + 6.10e27T^{2} \)
31 \( 1 + 1.04e14T + 2.16e28T^{2} \)
37 \( 1 + 1.69e14iT - 6.24e29T^{2} \)
41 \( 1 + 3.30e15T + 4.39e30T^{2} \)
43 \( 1 + 1.12e15iT - 1.08e31T^{2} \)
47 \( 1 - 3.49e15iT - 5.88e31T^{2} \)
53 \( 1 + 2.99e16iT - 5.77e32T^{2} \)
59 \( 1 + 5.83e16T + 4.42e33T^{2} \)
61 \( 1 - 2.33e16T + 8.34e33T^{2} \)
67 \( 1 + 2.05e17iT - 4.95e34T^{2} \)
71 \( 1 + 1.77e17T + 1.49e35T^{2} \)
73 \( 1 + 2.99e17iT - 2.53e35T^{2} \)
79 \( 1 - 9.22e16T + 1.13e36T^{2} \)
83 \( 1 + 1.20e18iT - 2.90e36T^{2} \)
89 \( 1 + 4.37e18T + 1.09e37T^{2} \)
97 \( 1 + 6.35e17iT - 5.60e37T^{2} \)
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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

−12.88456474472114353975502279865, −11.93439940968169456790696148851, −10.55769379278041864506134132987, −8.340069332482206578688293838856, −7.49906431701306562697036906915, −6.54738443691605908579520391151, −5.45324596898759018369576296933, −2.95878214732674949163294869575, −1.73634882752586563668290225406, −0.49820790553848793823951354309, 1.51244291385943638572918188276, 3.04474263412333404727129335586, 4.03287054661498612818662852287, 5.43641227986420585577582286195, 7.15203926375583865662927320267, 9.207382376219881786229597121137, 9.937977778585855813843514571167, 11.19174190144993960205409305858, 11.89437318975763521401208789359, 13.78484860642262508217495499165

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