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Properties

Label 2-60e2-100.79-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  + (−0.809 + 0.587i)5-s + (−1.11 + 0.363i)13-s + (−1.11 − 1.53i)17-s + (0.309 − 0.951i)25-s + (0.5 + 0.363i)29-s + (−1.80 + 0.587i)37-s + (−0.5 − 1.53i)41-s − 49-s + (0.690 − 0.951i)53-s + (0.5 − 1.53i)61-s + (0.690 − 0.951i)65-s + (−1.11 − 0.363i)73-s + (1.80 + 0.587i)85-s + (−0.190 + 0.587i)89-s + (1.11 − 1.53i)97-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)5-s + (−1.11 + 0.363i)13-s + (−1.11 − 1.53i)17-s + (0.309 − 0.951i)25-s + (0.5 + 0.363i)29-s + (−1.80 + 0.587i)37-s + (−0.5 − 1.53i)41-s − 49-s + (0.690 − 0.951i)53-s + (0.5 − 1.53i)61-s + (0.690 − 0.951i)65-s + (−1.11 − 0.363i)73-s + (1.80 + 0.587i)85-s + (−0.190 + 0.587i)89-s + (1.11 − 1.53i)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: $\chi_{3600} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ -0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3007659153\)
\(L(\frac12)\) \(\approx\) \(0.3007659153\)
\(L(1)\) 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 \( 1 + (0.809 - 0.587i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
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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

−8.489096472623870426321281234542, −7.57269548682059636916210637014, −6.93807734304246430322474004562, −6.62560500013083863187899628347, −5.18231871134800084587725311968, −4.74679029317903758268557891756, −3.74686920550808368791992454871, −2.89476261929471736936922618621, −2.05845295932628755071783506641, −0.16797970694903501799582829243, 1.48455746920464051305992032777, 2.62670045486089727583432503699, 3.69526012511733357314686300071, 4.42613765538212853670340185989, 5.07568582229331805525327098141, 6.00792228712597603146521484902, 6.89643654530348483581791284370, 7.56327484918996352767862311287, 8.392022637240776124690840892517, 8.725777797608876354712920324506

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