Properties

Label 2-60e2-15.14-c2-0-37
Degree $2$
Conductor $3600$
Sign $0.881 + 0.472i$
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  + i·7-s + 4.24i·11-s + 7i·13-s − 4.24·17-s − 7·19-s − 29.6·23-s − 29.6i·29-s − 17·31-s − 16i·37-s − 50.9i·41-s − 55i·43-s + 46.6·47-s + 48·49-s + 84.8·53-s + 55.1i·59-s + ⋯
L(s)  = 1  + 0.142i·7-s + 0.385i·11-s + 0.538i·13-s − 0.249·17-s − 0.368·19-s − 1.29·23-s − 1.02i·29-s − 0.548·31-s − 0.432i·37-s − 1.24i·41-s − 1.27i·43-s + 0.992·47-s + 0.979·49-s + 1.60·53-s + 0.934i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(98.0928\)
Root analytic conductor: \(9.90418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1),\ 0.881 + 0.472i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.653537321\)
\(L(\frac12)\) \(\approx\) \(1.653537321\)
\(L(2)\) 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 \( 1 - iT - 49T^{2} \)
11 \( 1 - 4.24iT - 121T^{2} \)
13 \( 1 - 7iT - 169T^{2} \)
17 \( 1 + 4.24T + 289T^{2} \)
19 \( 1 + 7T + 361T^{2} \)
23 \( 1 + 29.6T + 529T^{2} \)
29 \( 1 + 29.6iT - 841T^{2} \)
31 \( 1 + 17T + 961T^{2} \)
37 \( 1 + 16iT - 1.36e3T^{2} \)
41 \( 1 + 50.9iT - 1.68e3T^{2} \)
43 \( 1 + 55iT - 1.84e3T^{2} \)
47 \( 1 - 46.6T + 2.20e3T^{2} \)
53 \( 1 - 84.8T + 2.80e3T^{2} \)
59 \( 1 - 55.1iT - 3.48e3T^{2} \)
61 \( 1 - 65T + 3.72e3T^{2} \)
67 \( 1 - 49iT - 4.48e3T^{2} \)
71 \( 1 - 50.9iT - 5.04e3T^{2} \)
73 \( 1 - 88iT - 5.32e3T^{2} \)
79 \( 1 + 40T + 6.24e3T^{2} \)
83 \( 1 + 156.T + 6.88e3T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 - 41iT - 9.40e3T^{2} \)
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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

−8.605997402989778148866095046958, −7.34118781726713713358044458926, −7.08216726051260344056019456522, −5.91239658913113861251321831736, −5.51582434385837275262712927314, −4.18269426792325219441721130030, −3.98501714793386834719198359304, −2.49834403009239417562583158817, −1.92799397380963088445066090780, −0.48512447897100891555750904501, 0.73229931898736577015297534001, 1.89868580307932402102721322859, 2.91662175476413268462561485517, 3.77383342142740767800209859770, 4.59179299112211569662863766498, 5.50738586089344597633639852899, 6.16636741472943599436112381352, 6.95676102765559718176687214050, 7.78445595176875723064585921832, 8.384144099651074709846519790837

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