Properties

Label 2-5e2-1.1-c3-0-2
Degree $2$
Conductor $25$
Sign $-1$
Analytic cond. $1.47504$
Root an. cond. $1.21451$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 7·3-s − 7·4-s + 7·6-s − 6·7-s + 15·8-s + 22·9-s − 43·11-s + 49·12-s + 28·13-s + 6·14-s + 41·16-s − 91·17-s − 22·18-s − 35·19-s + 42·21-s + 43·22-s − 162·23-s − 105·24-s − 28·26-s + 35·27-s + 42·28-s + 160·29-s + 42·31-s − 161·32-s + 301·33-s + 91·34-s + ⋯
L(s)  = 1  − 0.353·2-s − 1.34·3-s − 7/8·4-s + 0.476·6-s − 0.323·7-s + 0.662·8-s + 0.814·9-s − 1.17·11-s + 1.17·12-s + 0.597·13-s + 0.114·14-s + 0.640·16-s − 1.29·17-s − 0.288·18-s − 0.422·19-s + 0.436·21-s + 0.416·22-s − 1.46·23-s − 0.893·24-s − 0.211·26-s + 0.249·27-s + 0.283·28-s + 1.02·29-s + 0.243·31-s − 0.889·32-s + 1.58·33-s + 0.459·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-1$
Analytic conductor: \(1.47504\)
Root analytic conductor: \(1.21451\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + T + p^{3} T^{2} \)
3 \( 1 + 7 T + p^{3} T^{2} \)
7 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 + 43 T + p^{3} T^{2} \)
13 \( 1 - 28 T + p^{3} T^{2} \)
17 \( 1 + 91 T + p^{3} T^{2} \)
19 \( 1 + 35 T + p^{3} T^{2} \)
23 \( 1 + 162 T + p^{3} T^{2} \)
29 \( 1 - 160 T + p^{3} T^{2} \)
31 \( 1 - 42 T + p^{3} T^{2} \)
37 \( 1 - 314 T + p^{3} T^{2} \)
41 \( 1 + 203 T + p^{3} T^{2} \)
43 \( 1 + 92 T + p^{3} T^{2} \)
47 \( 1 + 196 T + p^{3} T^{2} \)
53 \( 1 + 82 T + p^{3} T^{2} \)
59 \( 1 + 280 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 + 141 T + p^{3} T^{2} \)
71 \( 1 - 412 T + p^{3} T^{2} \)
73 \( 1 - 763 T + p^{3} T^{2} \)
79 \( 1 - 510 T + p^{3} T^{2} \)
83 \( 1 + 777 T + p^{3} T^{2} \)
89 \( 1 + 945 T + p^{3} T^{2} \)
97 \( 1 + 1246 T + p^{3} T^{2} \)
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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

−16.69961773350237799906472622064, −15.67093204476688056300260536922, −13.66966136312398800619567249168, −12.61757920584656200574446566405, −11.07012420934829748421911361250, −10.01294566678951525338184563642, −8.276840511944075708942991995655, −6.21043870300283489101869711821, −4.69045001633665639689119387631, 0, 4.69045001633665639689119387631, 6.21043870300283489101869711821, 8.276840511944075708942991995655, 10.01294566678951525338184563642, 11.07012420934829748421911361250, 12.61757920584656200574446566405, 13.66966136312398800619567249168, 15.67093204476688056300260536922, 16.69961773350237799906472622064

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