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Properties

Label 2-2541-231.86-c0-0-3
Degree $2$
Conductor $2541$
Sign $0.574 - 0.818i$
Analytic cond. $1.26812$
Root an. cond. $1.12611$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.104 − 0.994i)7-s + (0.669 + 0.743i)9-s + (−0.499 + 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (0.209 + 1.98i)19-s + (0.499 − 0.866i)21-s + (0.913 + 0.406i)25-s + (0.309 + 0.951i)27-s + (0.978 + 0.207i)28-s + (0.978 − 0.207i)31-s + (−0.809 + 0.587i)36-s + (−0.913 + 0.406i)37-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)3-s + (−0.104 + 0.994i)4-s + (0.104 − 0.994i)7-s + (0.669 + 0.743i)9-s + (−0.499 + 0.866i)12-s + (0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (0.209 + 1.98i)19-s + (0.499 − 0.866i)21-s + (0.913 + 0.406i)25-s + (0.309 + 0.951i)27-s + (0.978 + 0.207i)28-s + (0.978 − 0.207i)31-s + (−0.809 + 0.587i)36-s + (−0.913 + 0.406i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.574 - 0.818i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (1703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.574 - 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.686813494\)
\(L(\frac12)\) \(\approx\) \(1.686813494\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 - 0.406i)T \)
7 \( 1 + (-0.104 + 0.994i)T \)
11 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T^{2} \)
5 \( 1 + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (-0.209 - 1.98i)T + (-0.978 + 0.207i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.978 - 0.207i)T^{2} \)
53 \( 1 + (-0.913 + 0.406i)T^{2} \)
59 \( 1 + (0.978 + 0.207i)T^{2} \)
61 \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (1.33 + 1.48i)T + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.018861916770900954434567565015, −8.342155360196429723988347022527, −7.75172597652574514588672290392, −7.34207209049570794950238497611, −6.22834448696609269812756573534, −5.00250628473898255232055074935, −4.18121369534257294974783482957, −3.49192219940620170828346371872, −2.92578141685740708991646023027, −1.52005838600099981232640955989, 1.17841448927007039178935154618, 2.26300845188766402048960021602, 2.90327370804002604959320741411, 4.32077513330176904296663594526, 4.96284194177036215715306539562, 6.01508705524807674230635416543, 6.69826934020299744814480644630, 7.31553258190811477608035726181, 8.640735867767368909045972655456, 8.862419955186203125837921915610

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