Properties

Label 2-2541-1.1-c1-0-86
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.79·2-s − 3-s + 5.79·4-s + 3·5-s − 2.79·6-s − 7-s + 10.5·8-s + 9-s + 8.37·10-s − 5.79·12-s − 13-s − 2.79·14-s − 3·15-s + 17.9·16-s + 1.58·17-s + 2.79·18-s − 2.58·19-s + 17.3·20-s + 21-s + 3.58·23-s − 10.5·24-s + 4·25-s − 2.79·26-s − 27-s − 5.79·28-s − 10.1·29-s − 8.37·30-s + ⋯
L(s)  = 1  + 1.97·2-s − 0.577·3-s + 2.89·4-s + 1.34·5-s − 1.13·6-s − 0.377·7-s + 3.74·8-s + 0.333·9-s + 2.64·10-s − 1.67·12-s − 0.277·13-s − 0.746·14-s − 0.774·15-s + 4.48·16-s + 0.383·17-s + 0.657·18-s − 0.592·19-s + 3.88·20-s + 0.218·21-s + 0.747·23-s − 2.16·24-s + 0.800·25-s − 0.547·26-s − 0.192·27-s − 1.09·28-s − 1.88·29-s − 1.52·30-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: $\chi_{2541} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.934669386\)
\(L(\frac12)\) \(\approx\) \(6.934669386\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.79T + 2T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 1.58T + 17T^{2} \)
19 \( 1 + 2.58T + 19T^{2} \)
23 \( 1 - 3.58T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 + 5.58T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 7.16T + 41T^{2} \)
43 \( 1 - 7.58T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 0.417T + 53T^{2} \)
59 \( 1 + 4.58T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 0.582T + 67T^{2} \)
71 \( 1 + 7.16T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 2.41T + 83T^{2} \)
89 \( 1 + 9.16T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−9.105707724808017798853333694717, −7.52872696872587286192296732135, −7.02294558392243815964767589542, −6.09684599860608992804544130679, −5.76094621349309937392180384782, −5.14604717568885430800938250250, −4.25240172795480316619529293514, −3.33146277758417999291721434520, −2.34606150932660084426562243744, −1.56929116736556763104516901440, 1.56929116736556763104516901440, 2.34606150932660084426562243744, 3.33146277758417999291721434520, 4.25240172795480316619529293514, 5.14604717568885430800938250250, 5.76094621349309937392180384782, 6.09684599860608992804544130679, 7.02294558392243815964767589542, 7.52872696872587286192296732135, 9.105707724808017798853333694717

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