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Properties

Label 2-60e2-15.2-c1-0-24
Degree $2$
Conductor $3600$
Sign $-0.161 + 0.986i$
Analytic cond. $28.7461$
Root an. cond. $5.36154$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)7-s + 4.24i·11-s + (−3.67 + 3.67i)13-s + (−1.73 + 1.73i)17-s − 5i·19-s + (−1.73 − 1.73i)23-s + 4.24·29-s − 31-s + (2.44 + 2.44i)37-s − 8.48i·41-s + (1.22 − 1.22i)43-s + (5.19 − 5.19i)47-s − 4i·49-s + (−6.92 − 6.92i)53-s + 12.7·59-s + ⋯
L(s)  = 1  + (−0.462 − 0.462i)7-s + 1.27i·11-s + (−1.01 + 1.01i)13-s + (−0.420 + 0.420i)17-s − 1.14i·19-s + (−0.361 − 0.361i)23-s + 0.787·29-s − 0.179·31-s + (0.402 + 0.402i)37-s − 1.32i·41-s + (0.186 − 0.186i)43-s + (0.757 − 0.757i)47-s − 0.571i·49-s + (−0.951 − 0.951i)53-s + 1.65·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.161 + 0.986i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ -0.161 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8192701097\)
\(L(\frac12)\) \(\approx\) \(0.8192701097\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.22 + 1.22i)T + 7iT^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + (3.67 - 3.67i)T - 13iT^{2} \)
17 \( 1 + (1.73 - 1.73i)T - 17iT^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 + (1.73 + 1.73i)T + 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - 43iT^{2} \)
47 \( 1 + (-5.19 + 5.19i)T - 47iT^{2} \)
53 \( 1 + (6.92 + 6.92i)T + 53iT^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + (3.67 + 3.67i)T + 67iT^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + (-2.44 + 2.44i)T - 73iT^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (1.73 + 1.73i)T + 83iT^{2} \)
89 \( 1 - 8.48T + 89T^{2} \)
97 \( 1 + (8.57 + 8.57i)T + 97iT^{2} \)
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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

−8.396467896335181044449052120214, −7.33989495702606131725121590584, −6.96293906478682188495936437551, −6.36547229758547821521914846460, −5.10466135218924981412236864354, −4.53199763606888414720546276612, −3.82573558280879248791473246384, −2.56152219494541457428512649476, −1.88610740943763290888904766175, −0.26549009970040889200333098656, 1.04928644689427286756683753151, 2.56466152726544929201166502995, 3.07558765993966612196336496705, 4.08437726024321589206627924797, 5.10503087976545631962719789781, 5.86016107490960807002687749494, 6.30215133462146722476269621147, 7.44210913926205108434209692176, 7.997203446723132876773537828066, 8.711551099985230361218716971526

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