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Properties

Label 2-60e2-20.19-c0-0-1
Degree $2$
Conductor $3600$
Sign $0.447 - 0.894i$
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  + 2i·13-s + 2i·37-s − 49-s + 2·61-s + 2i·73-s − 2i·97-s + 2·109-s + ⋯
L(s)  = 1  + 2i·13-s + 2i·37-s − 49-s + 2·61-s + 2i·73-s − 2i·97-s + 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.160497482\)
\(L(\frac12)\) \(\approx\) \(1.160497482\)
\(L(1)\) 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 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 2iT - T^{2} \)
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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

−8.744639515276015546466439988202, −8.307981379522438899276826772171, −7.20092763799726516429568611009, −6.72554817250857957810675502649, −5.99549310123490890568198290167, −4.93339366880960330945309581361, −4.33147646517080615904703448315, −3.44680689451704994952120901457, −2.33871751472027258131116253731, −1.41611566124676857499487025226, 0.70459278303312580829601908912, 2.13787882853803618420527634141, 3.10713449069616854991201655439, 3.82968293561725444323099180556, 4.95386383490508981866941805021, 5.56888400453275466853854184271, 6.27155627601672985441647049691, 7.28223998137666742376528928606, 7.85287447217424325619411767762, 8.507216344326230535307555964467

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