Properties

Label 2-60e2-300.23-c0-0-1
Degree $2$
Conductor $3600$
Sign $0.675 + 0.737i$
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  + (−0.891 − 0.453i)5-s + (0.142 + 0.896i)13-s + (0.734 − 1.44i)17-s + (0.587 + 0.809i)25-s + (0.0966 − 0.297i)29-s + (−0.309 + 0.0489i)37-s + (0.533 − 0.734i)41-s i·49-s + (−0.280 − 0.550i)53-s + (1.53 − 1.11i)61-s + (0.280 − 0.863i)65-s + (1.76 + 0.278i)73-s + (−1.30 + 0.951i)85-s + (1.59 − 1.16i)89-s + (−0.896 − 1.76i)97-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)5-s + (0.142 + 0.896i)13-s + (0.734 − 1.44i)17-s + (0.587 + 0.809i)25-s + (0.0966 − 0.297i)29-s + (−0.309 + 0.0489i)37-s + (0.533 − 0.734i)41-s i·49-s + (−0.280 − 0.550i)53-s + (1.53 − 1.11i)61-s + (0.280 − 0.863i)65-s + (1.76 + 0.278i)73-s + (−1.30 + 0.951i)85-s + (1.59 − 1.16i)89-s + (−0.896 − 1.76i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.675 + 0.737i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (3023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.675 + 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.038323649\)
\(L(\frac12)\) \(\approx\) \(1.038323649\)
\(L(1)\) 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 \( 1 + (0.891 + 0.453i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.734 + 1.44i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.0966 + 0.297i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.280 + 0.550i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (-1.59 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \)
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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

−8.600362995951573995479272816106, −7.900248868954383470748883081983, −7.19615765648904113584224126906, −6.59818709758836535033791931756, −5.42718957846115072003719631262, −4.85188153551205455069845242537, −3.98041917999723148351395345492, −3.26171681890443191568351265737, −2.08896588780027101286253778515, −0.73293158763106898779783793541, 1.12747058406160642558819075345, 2.55083874842322844418643324634, 3.45223840438058805880082510426, 4.02162037068275160900945817177, 5.04453938162711661795476470647, 5.91932988608958341518442558800, 6.59681238637636431393394734477, 7.53908006766832770996442794512, 8.021534251378567756853660791092, 8.597483381082438061579901515205

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