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Properties

Label 2-60e2-100.71-c0-0-0
Degree $2$
Conductor $3600$
Sign $0.637 - 0.770i$
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  + (−0.587 − 0.809i)5-s + (0.5 + 1.53i)13-s + (−1.53 + 1.11i)17-s + (−0.309 + 0.951i)25-s + (1.53 + 1.11i)29-s + (0.190 + 0.587i)37-s + (−0.363 − 1.11i)41-s + 49-s + (0.951 + 0.690i)53-s + (−0.5 + 1.53i)61-s + (0.951 − 1.30i)65-s + (0.5 − 1.53i)73-s + (1.80 + 0.587i)85-s + (0.587 − 1.80i)89-s + (0.5 + 0.363i)97-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)5-s + (0.5 + 1.53i)13-s + (−1.53 + 1.11i)17-s + (−0.309 + 0.951i)25-s + (1.53 + 1.11i)29-s + (0.190 + 0.587i)37-s + (−0.363 − 1.11i)41-s + 49-s + (0.951 + 0.690i)53-s + (−0.5 + 1.53i)61-s + (0.951 − 1.30i)65-s + (0.5 − 1.53i)73-s + (1.80 + 0.587i)85-s + (0.587 − 1.80i)89-s + (0.5 + 0.363i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9769324037\)
\(L(\frac12)\) \(\approx\) \(0.9769324037\)
\(L(1)\) 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 \( 1 + (0.587 + 0.809i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.839761778738913035127276393164, −8.354574134813022895562884084135, −7.29148400599101576827203786956, −6.65825757890541619020609830782, −5.92636611596063433598161724285, −4.78463276196012089990374896051, −4.31976625278596687137688498718, −3.59299710231483935109292300067, −2.23338534528102247341114726660, −1.27740266124191497122101175838, 0.61624336774736375767605281258, 2.41696449527133089787412493554, 2.97967072308346940758627824423, 3.95768038881928640272438672718, 4.74394645691854679892025326167, 5.69434680655790254656780212086, 6.56124048560841812021505860694, 7.05421462757863328635071053467, 8.048404637446572872353621932820, 8.335736218022756517968239833606

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