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Properties

Label 2-60e2-3.2-c0-0-2
Degree $2$
Conductor $3600$
Sign $-0.577 + 0.816i$
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  − 7-s + 1.41i·11-s − 13-s − 1.41i·17-s − 19-s − 1.41i·23-s − 1.41i·29-s − 31-s + 43-s − 1.41i·47-s − 1.41i·59-s + 61-s − 67-s − 1.41i·77-s + 1.41i·83-s + ⋯
L(s)  = 1  − 7-s + 1.41i·11-s − 13-s − 1.41i·17-s − 19-s − 1.41i·23-s − 1.41i·29-s − 31-s + 43-s − 1.41i·47-s − 1.41i·59-s + 61-s − 67-s − 1.41i·77-s + 1.41i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4404299148\)
\(L(\frac12)\) \(\approx\) \(0.4404299148\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.504376287316791111261920639139, −7.57639428319313693961799440129, −6.94056524709176694929115734238, −6.48155472651012750334871403465, −5.37459738163883289443872443183, −4.62010587423756121112540391670, −3.93289219689489016547639791972, −2.64698171673129953045298894380, −2.19428170232940965820214074653, −0.24081552773675493663994565581, 1.49527977101021543001190909688, 2.77367984686769122102100784281, 3.48759906384755615567839740278, 4.23370698798162955198106245853, 5.48529931616806908716620456779, 5.93662682883774945513389250973, 6.72155611191140098040230350033, 7.49528633884229337316987183669, 8.304910865210044839633835317920, 9.043086983049464747135441798559

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