Properties

Label 2-25751-1.1-c1-0-0
Degree $2$
Conductor $25751$
Sign $-1$
Analytic cond. $205.622$
Root an. cond. $14.3395$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 4-s − 2·5-s + 2·6-s − 3·7-s + 3·8-s + 9-s + 2·10-s − 11-s + 2·12-s − 5·13-s + 3·14-s + 4·15-s − 16-s − 6·17-s − 18-s − 8·19-s + 2·20-s + 6·21-s + 22-s − 6·23-s − 6·24-s − 25-s + 5·26-s + 4·27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s − 1.38·13-s + 0.801·14-s + 1.03·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s + 0.447·20-s + 1.30·21-s + 0.213·22-s − 1.25·23-s − 1.22·24-s − 1/5·25-s + 0.980·26-s + 0.769·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25751 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25751\)    =    \(11 \cdot 2341\)
Sign: $-1$
Analytic conductor: \(205.622\)
Root analytic conductor: \(14.3395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 25751,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
2341 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.27345958720139, −15.80500951569142, −15.20356733659090, −14.70735679103314, −13.95204990042930, −13.16007963727968, −12.81215670353112, −12.44967202510566, −11.66344722363233, −11.29379866845009, −10.61024625810045, −10.20350550820871, −9.690225918248316, −9.028161852247612, −8.473127061525397, −7.919261517409008, −7.137818026996893, −6.796464044359082, −6.085222378745268, −5.408268920980588, −4.721290107802177, −4.168603075328746, −3.668028121016701, −2.497951991107593, −1.775878884561265, 0, 0, 0, 1.775878884561265, 2.497951991107593, 3.668028121016701, 4.168603075328746, 4.721290107802177, 5.408268920980588, 6.085222378745268, 6.796464044359082, 7.137818026996893, 7.919261517409008, 8.473127061525397, 9.028161852247612, 9.690225918248316, 10.20350550820871, 10.61024625810045, 11.29379866845009, 11.66344722363233, 12.44967202510566, 12.81215670353112, 13.16007963727968, 13.95204990042930, 14.70735679103314, 15.20356733659090, 15.80500951569142, 16.27345958720139

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