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Properties

Label 2-60e2-5.3-c0-0-2
Degree $2$
Conductor $3600$
Sign $-0.608 + 0.793i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  + (−1.22 + 1.22i)7-s + (−1.22 − 1.22i)13-s − i·19-s − 31-s + (−1.22 − 1.22i)43-s − 1.99i·49-s − 61-s + (1.22 − 1.22i)67-s − 2i·79-s + 2.99·91-s + (−1.22 + 1.22i)97-s − i·109-s + ⋯
L(s)  = 1  + (−1.22 + 1.22i)7-s + (−1.22 − 1.22i)13-s − i·19-s − 31-s + (−1.22 − 1.22i)43-s − 1.99i·49-s − 61-s + (1.22 − 1.22i)67-s − 2i·79-s + 2.99·91-s + (−1.22 + 1.22i)97-s − i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ -0.608 + 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3414570620\)
\(L(\frac12)\) \(\approx\) \(0.3414570620\)
\(L(1)\) 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 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
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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

−8.611795953374382749983518073604, −7.68707915555352933615119637245, −6.96169067137254149454552360070, −6.21486185534567738050643774781, −5.41423435235535838430389359661, −4.92237579562580777726862184890, −3.53643820291953603802348488392, −2.88096029348874040358272467948, −2.14658291800270743101589395992, −0.18709629992634672547776406917, 1.48412636655938604056864027190, 2.68553911741543721681178171093, 3.68051551916732497067708091023, 4.22831217964985383981072381300, 5.17102365277129305154966812557, 6.22094241439894432794194788668, 6.85739674258892575170937522114, 7.33445598941504071616220216361, 8.147060563538888671658238111903, 9.210487460867424970830655556009

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