Properties

Label 2-5e2-1.1-c11-0-2
Degree $2$
Conductor $25$
Sign $1$
Analytic cond. $19.2085$
Root an. cond. $4.38275$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·2-s − 252·3-s − 1.47e3·4-s − 6.04e3·6-s + 1.67e4·7-s − 8.44e4·8-s − 1.13e5·9-s + 5.34e5·11-s + 3.70e5·12-s + 5.77e5·13-s + 4.01e5·14-s + 9.87e5·16-s + 6.90e6·17-s − 2.72e6·18-s + 1.06e7·19-s − 4.21e6·21-s + 1.28e7·22-s − 1.86e7·23-s + 2.12e7·24-s + 1.38e7·26-s + 7.32e7·27-s − 2.46e7·28-s + 1.28e8·29-s − 5.28e7·31-s + 1.96e8·32-s − 1.34e8·33-s + 1.65e8·34-s + ⋯
L(s)  = 1  + 0.530·2-s − 0.598·3-s − 0.718·4-s − 0.317·6-s + 0.376·7-s − 0.911·8-s − 0.641·9-s + 1.00·11-s + 0.430·12-s + 0.431·13-s + 0.199·14-s + 0.235·16-s + 1.17·17-s − 0.340·18-s + 0.987·19-s − 0.225·21-s + 0.530·22-s − 0.603·23-s + 0.545·24-s + 0.228·26-s + 0.982·27-s − 0.270·28-s + 1.16·29-s − 0.331·31-s + 1.03·32-s − 0.599·33-s + 0.625·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $1$
Analytic conductor: \(19.2085\)
Root analytic conductor: \(4.38275\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(1.632375257\)
\(L(\frac12)\) \(\approx\) \(1.632375257\)
\(L(\frac{13}{2})\) 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 \( 1 - 3 p^{3} T + p^{11} T^{2} \)
3 \( 1 + 28 p^{2} T + p^{11} T^{2} \)
7 \( 1 - 2392 p T + p^{11} T^{2} \)
11 \( 1 - 534612 T + p^{11} T^{2} \)
13 \( 1 - 577738 T + p^{11} T^{2} \)
17 \( 1 - 6905934 T + p^{11} T^{2} \)
19 \( 1 - 10661420 T + p^{11} T^{2} \)
23 \( 1 + 18643272 T + p^{11} T^{2} \)
29 \( 1 - 128406630 T + p^{11} T^{2} \)
31 \( 1 + 52843168 T + p^{11} T^{2} \)
37 \( 1 - 182213314 T + p^{11} T^{2} \)
41 \( 1 - 308120442 T + p^{11} T^{2} \)
43 \( 1 - 17125708 T + p^{11} T^{2} \)
47 \( 1 + 2687348496 T + p^{11} T^{2} \)
53 \( 1 - 1596055698 T + p^{11} T^{2} \)
59 \( 1 + 5189203740 T + p^{11} T^{2} \)
61 \( 1 - 6956478662 T + p^{11} T^{2} \)
67 \( 1 - 15481826884 T + p^{11} T^{2} \)
71 \( 1 - 9791485272 T + p^{11} T^{2} \)
73 \( 1 + 1463791322 T + p^{11} T^{2} \)
79 \( 1 - 38116845680 T + p^{11} T^{2} \)
83 \( 1 - 29335099668 T + p^{11} T^{2} \)
89 \( 1 + 24992917110 T + p^{11} T^{2} \)
97 \( 1 + 75013568546 T + p^{11} T^{2} \)
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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

−14.58953831482676741852927654319, −13.92517079548481304513552363366, −12.33692344425067775161222860231, −11.45227418950418789076071804759, −9.687987714091356281872312562067, −8.276755585448261435239526991938, −6.16895544016659072089894759507, −5.01850388149551306184008702080, −3.46024287685394911500515172135, −0.887582569880732194448271490195, 0.887582569880732194448271490195, 3.46024287685394911500515172135, 5.01850388149551306184008702080, 6.16895544016659072089894759507, 8.276755585448261435239526991938, 9.687987714091356281872312562067, 11.45227418950418789076071804759, 12.33692344425067775161222860231, 13.92517079548481304513552363366, 14.58953831482676741852927654319

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