Properties

Label 2-1200-15.14-c2-0-64
Degree $2$
Conductor $1200$
Sign $-0.929 - 0.368i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.23 − 2i)3-s + 2i·7-s + (1.00 − 8.94i)9-s + 13.4i·11-s + 8i·13-s − 13.4·17-s − 34·19-s + (4 + 4.47i)21-s − 40.2·23-s + (−15.6 − 22.0i)27-s − 40.2i·29-s − 14·31-s + (26.8 + 30.0i)33-s − 56i·37-s + (16 + 17.8i)39-s + ⋯
L(s)  = 1  + (0.745 − 0.666i)3-s + 0.285i·7-s + (0.111 − 0.993i)9-s + 1.21i·11-s + 0.615i·13-s − 0.789·17-s − 1.78·19-s + (0.190 + 0.212i)21-s − 1.74·23-s + (−0.579 − 0.814i)27-s − 1.38i·29-s − 0.451·31-s + (0.813 + 0.909i)33-s − 1.51i·37-s + (0.410 + 0.458i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.929 - 0.368i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01056876983\)
\(L(\frac12)\) \(\approx\) \(0.01056876983\)
\(L(2)\) 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 \( 1 + (-2.23 + 2i)T \)
5 \( 1 \)
good7 \( 1 - 2iT - 49T^{2} \)
11 \( 1 - 13.4iT - 121T^{2} \)
13 \( 1 - 8iT - 169T^{2} \)
17 \( 1 + 13.4T + 289T^{2} \)
19 \( 1 + 34T + 361T^{2} \)
23 \( 1 + 40.2T + 529T^{2} \)
29 \( 1 + 40.2iT - 841T^{2} \)
31 \( 1 + 14T + 961T^{2} \)
37 \( 1 + 56iT - 1.36e3T^{2} \)
41 \( 1 - 26.8iT - 1.68e3T^{2} \)
43 \( 1 + 8iT - 1.84e3T^{2} \)
47 \( 1 + 40.2T + 2.20e3T^{2} \)
53 \( 1 + 40.2T + 2.80e3T^{2} \)
59 \( 1 + 13.4iT - 3.48e3T^{2} \)
61 \( 1 + 46T + 3.72e3T^{2} \)
67 \( 1 - 32iT - 4.48e3T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 106iT - 5.32e3T^{2} \)
79 \( 1 + 22T + 6.24e3T^{2} \)
83 \( 1 - 120.T + 6.88e3T^{2} \)
89 \( 1 - 107. iT - 7.92e3T^{2} \)
97 \( 1 + 122iT - 9.40e3T^{2} \)
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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

−9.079774559481353912208652494311, −8.233002992134917170301318635293, −7.54683543673503006692789183266, −6.61146285672673865505168395977, −6.05349648417671664937376774984, −4.48244269934000413310430976965, −3.91128773101413937118617238929, −2.22765364733967551678613987539, −2.00793518363584693449368993657, −0.00247494265833574726372291768, 1.86220032734003957367456191315, 3.01370392120000591473021930520, 3.86263442570371877847876657115, 4.69707560066858465016458093595, 5.79720231659006233478279235949, 6.66638928993202976700456397516, 7.88602232359321335209910843962, 8.456007467533345531558320566434, 9.027989872721636658977031575240, 10.12157897081506421695098477130

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