Properties

Label 2-60e2-3.2-c2-0-50
Degree $2$
Conductor $3600$
Sign $-0.577 + 0.816i$
Analytic cond. $98.0928$
Root an. cond. $9.90418$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.4·7-s − 17.6i·11-s + 7.48·13-s + 16.9i·17-s + 10.9·19-s + 21.9i·23-s + 47.3i·29-s + 16.9·31-s + 5.53·37-s − 66.3i·41-s + 38.9·43-s − 32.5i·47-s + 132.·49-s + 11.2i·53-s − 31.8i·59-s + ⋯
L(s)  = 1  − 1.92·7-s − 1.60i·11-s + 0.575·13-s + 0.998i·17-s + 0.577·19-s + 0.952i·23-s + 1.63i·29-s + 0.547·31-s + 0.149·37-s − 1.61i·41-s + 0.906·43-s − 0.692i·47-s + 2.71·49-s + 0.212i·53-s − 0.540i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(98.0928\)
Root analytic conductor: \(9.90418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7512593356\)
\(L(\frac12)\) \(\approx\) \(0.7512593356\)
\(L(2)\) 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 \( 1 + 13.4T + 49T^{2} \)
11 \( 1 + 17.6iT - 121T^{2} \)
13 \( 1 - 7.48T + 169T^{2} \)
17 \( 1 - 16.9iT - 289T^{2} \)
19 \( 1 - 10.9T + 361T^{2} \)
23 \( 1 - 21.9iT - 529T^{2} \)
29 \( 1 - 47.3iT - 841T^{2} \)
31 \( 1 - 16.9T + 961T^{2} \)
37 \( 1 - 5.53T + 1.36e3T^{2} \)
41 \( 1 + 66.3iT - 1.68e3T^{2} \)
43 \( 1 - 38.9T + 1.84e3T^{2} \)
47 \( 1 + 32.5iT - 2.20e3T^{2} \)
53 \( 1 - 11.2iT - 2.80e3T^{2} \)
59 \( 1 + 31.8iT - 3.48e3T^{2} \)
61 \( 1 - 46.9T + 3.72e3T^{2} \)
67 \( 1 + 76T + 4.48e3T^{2} \)
71 \( 1 + 77.7iT - 5.04e3T^{2} \)
73 \( 1 + 94.9T + 5.32e3T^{2} \)
79 \( 1 - 6.92T + 6.24e3T^{2} \)
83 \( 1 - 62.1iT - 6.88e3T^{2} \)
89 \( 1 - 62.2iT - 7.92e3T^{2} \)
97 \( 1 - 124.T + 9.40e3T^{2} \)
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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

−8.251705181249109389491146525897, −7.25118556465533541424495408827, −6.54174735284730820411477904774, −5.89981147808746665647165110752, −5.43182518324316677362814973518, −3.83006857001908826573049304447, −3.48988079727488196726960259151, −2.76842498517207105295229672494, −1.24989833817534671668105779135, −0.20431381529967858563518611151, 0.923912186432639386413852387468, 2.44475434725568436289809054309, 2.97632856003991694980726257195, 4.06965983306130057801523154073, 4.67883655695569554359389073053, 5.88074199798297924241705593697, 6.41139811423753346276701977881, 7.11219501990289888416038314478, 7.70735195610107693022623751964, 8.812886665025206908851589870042

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