Properties

Label 2-2541-1.1-c1-0-103
Degree $2$
Conductor $2541$
Sign $-1$
Analytic cond. $20.2899$
Root an. cond. $4.50444$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.39·2-s − 3-s + 3.71·4-s − 2.58·5-s − 2.39·6-s + 7-s + 4.09·8-s + 9-s − 6.19·10-s − 3.71·12-s − 6.78·13-s + 2.39·14-s + 2.58·15-s + 2.36·16-s + 3.14·17-s + 2.39·18-s + 1.04·19-s − 9.61·20-s − 21-s − 6.52·23-s − 4.09·24-s + 1.70·25-s − 16.2·26-s − 27-s + 3.71·28-s + 0.607·29-s + 6.19·30-s + ⋯
L(s)  = 1  + 1.69·2-s − 0.577·3-s + 1.85·4-s − 1.15·5-s − 0.975·6-s + 0.377·7-s + 1.44·8-s + 0.333·9-s − 1.95·10-s − 1.07·12-s − 1.88·13-s + 0.638·14-s + 0.668·15-s + 0.590·16-s + 0.762·17-s + 0.563·18-s + 0.239·19-s − 2.15·20-s − 0.218·21-s − 1.36·23-s − 0.835·24-s + 0.341·25-s − 3.17·26-s − 0.192·27-s + 0.701·28-s + 0.112·29-s + 1.13·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(20.2899\)
Root analytic conductor: \(4.50444\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2541,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - 2.39T + 2T^{2} \)
5 \( 1 + 2.58T + 5T^{2} \)
13 \( 1 + 6.78T + 13T^{2} \)
17 \( 1 - 3.14T + 17T^{2} \)
19 \( 1 - 1.04T + 19T^{2} \)
23 \( 1 + 6.52T + 23T^{2} \)
29 \( 1 - 0.607T + 29T^{2} \)
31 \( 1 + 8.83T + 31T^{2} \)
37 \( 1 - 8.94T + 37T^{2} \)
41 \( 1 + 8.69T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 - 1.89T + 53T^{2} \)
59 \( 1 + 0.174T + 59T^{2} \)
61 \( 1 + 8.13T + 61T^{2} \)
67 \( 1 + 7.33T + 67T^{2} \)
71 \( 1 - 2.70T + 71T^{2} \)
73 \( 1 + 0.589T + 73T^{2} \)
79 \( 1 - 9.80T + 79T^{2} \)
83 \( 1 + 8.74T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−8.043996957413836265728153860486, −7.49810974215811869086182876393, −6.92150887111400690683042950591, −5.86193134440943465279899268867, −5.23284313817329818224790798229, −4.53631469287683408170350038017, −3.93224392094103693532682784349, −3.03562273768791545050073872595, −1.93606233753585539488677769887, 0, 1.93606233753585539488677769887, 3.03562273768791545050073872595, 3.93224392094103693532682784349, 4.53631469287683408170350038017, 5.23284313817329818224790798229, 5.86193134440943465279899268867, 6.92150887111400690683042950591, 7.49810974215811869086182876393, 8.043996957413836265728153860486

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