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Properties

Label 2-2541-21.11-c0-0-1
Degree $2$
Conductor $2541$
Sign $0.0633 - 0.997i$
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  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + 13-s + (−0.499 − 0.866i)16-s + (1 + 1.73i)19-s + (0.499 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.499 + 0.866i)28-s + (0.5 − 0.866i)31-s + 0.999·36-s + (0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + 13-s + (−0.499 − 0.866i)16-s + (1 + 1.73i)19-s + (0.499 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.499 + 0.866i)28-s + (0.5 − 0.866i)31-s + 0.999·36-s + (0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (2300, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9614306166\)
\(L(\frac12)\) \(\approx\) \(0.9614306166\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.439543618817687144126469835382, −8.381993257289067165455865986824, −7.942877846003719602772132456213, −7.08492906487025555493090533879, −5.99042005758644570158940532015, −5.30566183967953850175850123452, −4.23086597230805776665375564311, −3.89847378477002643164069590218, −3.07135451202125401810209076042, −1.21766498507176773728112996384, 0.843402172575182999960874410668, 1.87807375781189837977175512605, 2.89552595622373763303243230717, 4.46783358296685124686944396522, 5.11579317207908107271756250351, 5.88350100019413366994995819911, 6.35777602824250233356988168181, 7.33782981579372398610922781935, 8.199364225079009780904544783172, 8.931506610858884151377457095874

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