Properties

Label 2-5e2-1.1-c7-0-7
Degree $2$
Conductor $25$
Sign $-1$
Analytic cond. $7.80962$
Root an. cond. $2.79457$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.23·2-s + 5.47·3-s − 100.·4-s + 28.6·6-s − 769.·7-s − 1.19e3·8-s − 2.15e3·9-s + 5.35e3·11-s − 550.·12-s − 1.40e4·13-s − 4.03e3·14-s + 6.60e3·16-s − 1.24e4·17-s − 1.12e4·18-s − 5.03e3·19-s − 4.21e3·21-s + 2.80e4·22-s + 7.51e4·23-s − 6.55e3·24-s − 7.35e4·26-s − 2.37e4·27-s + 7.74e4·28-s + 1.95e5·29-s − 9.35e4·31-s + 1.87e5·32-s + 2.93e4·33-s − 6.49e4·34-s + ⋯
L(s)  = 1  + 0.462·2-s + 0.117·3-s − 0.785·4-s + 0.0542·6-s − 0.848·7-s − 0.826·8-s − 0.986·9-s + 1.21·11-s − 0.0919·12-s − 1.77·13-s − 0.392·14-s + 0.402·16-s − 0.612·17-s − 0.456·18-s − 0.168·19-s − 0.0993·21-s + 0.561·22-s + 1.28·23-s − 0.0967·24-s − 0.820·26-s − 0.232·27-s + 0.666·28-s + 1.48·29-s − 0.564·31-s + 1.01·32-s + 0.142·33-s − 0.283·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-1$
Analytic conductor: \(7.80962\)
Root analytic conductor: \(2.79457\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 - 5.23T + 128T^{2} \)
3 \( 1 - 5.47T + 2.18e3T^{2} \)
7 \( 1 + 769.T + 8.23e5T^{2} \)
11 \( 1 - 5.35e3T + 1.94e7T^{2} \)
13 \( 1 + 1.40e4T + 6.27e7T^{2} \)
17 \( 1 + 1.24e4T + 4.10e8T^{2} \)
19 \( 1 + 5.03e3T + 8.93e8T^{2} \)
23 \( 1 - 7.51e4T + 3.40e9T^{2} \)
29 \( 1 - 1.95e5T + 1.72e10T^{2} \)
31 \( 1 + 9.35e4T + 2.75e10T^{2} \)
37 \( 1 + 1.61e5T + 9.49e10T^{2} \)
41 \( 1 + 2.87e4T + 1.94e11T^{2} \)
43 \( 1 + 7.39e5T + 2.71e11T^{2} \)
47 \( 1 + 1.06e6T + 5.06e11T^{2} \)
53 \( 1 + 6.26e5T + 1.17e12T^{2} \)
59 \( 1 - 2.14e6T + 2.48e12T^{2} \)
61 \( 1 + 2.57e6T + 3.14e12T^{2} \)
67 \( 1 - 8.07e5T + 6.06e12T^{2} \)
71 \( 1 + 1.72e6T + 9.09e12T^{2} \)
73 \( 1 - 1.74e6T + 1.10e13T^{2} \)
79 \( 1 + 2.46e6T + 1.92e13T^{2} \)
83 \( 1 + 6.90e6T + 2.71e13T^{2} \)
89 \( 1 - 2.63e6T + 4.42e13T^{2} \)
97 \( 1 - 1.01e7T + 8.07e13T^{2} \)
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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

−14.95937024719211189959730435798, −14.18663853440256213057478113455, −12.88966781965988083640468954127, −11.79467104745418520971479149715, −9.752193847258098494785779734218, −8.748632304742897274543466814240, −6.60748954782632877725163608559, −4.88220612848529370419279847640, −3.12223906792142027095375231614, 0, 3.12223906792142027095375231614, 4.88220612848529370419279847640, 6.60748954782632877725163608559, 8.748632304742897274543466814240, 9.752193847258098494785779734218, 11.79467104745418520971479149715, 12.88966781965988083640468954127, 14.18663853440256213057478113455, 14.95937024719211189959730435798

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