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Properties

Label 2-2541-231.62-c0-0-2
Degree $2$
Conductor $2541$
Sign $0.852 - 0.522i$
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  + (0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.453 + 0.891i)7-s + (0.809 + 0.587i)9-s − 0.999i·12-s + (1.14 + 0.831i)13-s + (−0.809 + 0.587i)16-s + (−0.437 + 1.34i)19-s + (−0.707 + 0.707i)21-s + (−0.309 + 0.951i)25-s + (0.587 + 0.809i)27-s + (0.987 + 0.156i)28-s + (1.17 − 1.61i)31-s + (0.309 − 0.951i)36-s + (0.831 + 1.14i)39-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.453 + 0.891i)7-s + (0.809 + 0.587i)9-s − 0.999i·12-s + (1.14 + 0.831i)13-s + (−0.809 + 0.587i)16-s + (−0.437 + 1.34i)19-s + (−0.707 + 0.707i)21-s + (−0.309 + 0.951i)25-s + (0.587 + 0.809i)27-s + (0.987 + 0.156i)28-s + (1.17 − 1.61i)31-s + (0.309 − 0.951i)36-s + (0.831 + 1.14i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.852 - 0.522i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.542834307\)
\(L(\frac12)\) \(\approx\) \(1.542834307\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (0.453 - 0.891i)T \)
11 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-1.14 - 0.831i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-1.17 + 1.61i)T + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-1.14 + 0.831i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.437 + 1.34i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.144025768442538138301531557499, −8.610551195622005110829810825155, −7.895398989742428742826658696784, −6.69385663579823259329066430172, −6.03464039022321176804186078251, −5.31344147983626256097843656673, −4.20011577900202531397693041113, −3.62045277530159709778451881062, −2.33589200176059110215086492411, −1.56862236745003424117683346125, 1.01735659073506865141466525930, 2.66361277882754619603889858150, 3.22381975631720667408035792657, 4.05710906713776130770076569679, 4.72156893188074052580605805346, 6.26648415920309959731809093764, 6.90338979293341027943523693309, 7.58108864272716301076320048712, 8.478354145159723420638114857692, 8.606175671426679845742066677468

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