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Properties

Label 2-60e2-5.2-c0-0-1
Degree $2$
Conductor $3600$
Sign $0.899 - 0.437i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564348081\)
\(L(\frac12)\) \(\approx\) \(1.564348081\)
\(L(1)\) 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

−8.765572515601009497378699240127, −8.010698278072573663734382271265, −7.64866312719275285820557229726, −6.34722756512023033785902435202, −5.61892121981556677845251171144, −5.29077559139149607011720135358, −4.15314666915864503242565834440, −3.27417903871634998758329991474, −2.23192356955134050998751645406, −1.33137812937374454029757416550, 1.12441530338171008108935494728, 1.92725230633632622126262153481, 3.29550047965116642069102196336, 4.32467596511437546915187135954, 4.53593743802768765844854215197, 5.69590324476150073284744629483, 6.55904361689522022197439415846, 7.29855106198938191845566963028, 7.81499910766011028219031007048, 8.770873889047019208812981511778

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