Properties

Label 2-1200-3.2-c2-0-27
Degree $2$
Conductor $1200$
Sign $-0.166 - 0.986i$
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  + (0.5 + 2.95i)3-s + 8·7-s + (−8.5 + 2.95i)9-s − 17.7i·11-s − 2·13-s + 17.7i·17-s − 11·19-s + (4 + 23.6i)21-s + 35.4i·23-s + (−13 − 23.6i)27-s + 35.4i·29-s + 46·31-s + (52.5 − 8.87i)33-s + 16·37-s + (−1 − 5.91i)39-s + ⋯
L(s)  = 1  + (0.166 + 0.986i)3-s + 1.14·7-s + (−0.944 + 0.328i)9-s − 1.61i·11-s − 0.153·13-s + 1.04i·17-s − 0.578·19-s + (0.190 + 1.12i)21-s + 1.54i·23-s + (−0.481 − 0.876i)27-s + 1.22i·29-s + 1.48·31-s + (1.59 − 0.268i)33-s + 0.432·37-s + (−0.0256 − 0.151i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.166 - 0.986i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.166 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.044046289\)
\(L(\frac12)\) \(\approx\) \(2.044046289\)
\(L(2)\) 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 \( 1 + (-0.5 - 2.95i)T \)
5 \( 1 \)
good7 \( 1 - 8T + 49T^{2} \)
11 \( 1 + 17.7iT - 121T^{2} \)
13 \( 1 + 2T + 169T^{2} \)
17 \( 1 - 17.7iT - 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 - 35.4iT - 529T^{2} \)
29 \( 1 - 35.4iT - 841T^{2} \)
31 \( 1 - 46T + 961T^{2} \)
37 \( 1 - 16T + 1.36e3T^{2} \)
41 \( 1 - 53.2iT - 1.68e3T^{2} \)
43 \( 1 - 62T + 1.84e3T^{2} \)
47 \( 1 + 35.4iT - 2.20e3T^{2} \)
53 \( 1 - 35.4iT - 2.80e3T^{2} \)
59 \( 1 - 70.9iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 - 113T + 4.48e3T^{2} \)
71 \( 1 - 106. iT - 5.04e3T^{2} \)
73 \( 1 + 101T + 5.32e3T^{2} \)
79 \( 1 + 68T + 6.24e3T^{2} \)
83 \( 1 + 17.7iT - 6.88e3T^{2} \)
89 \( 1 - 53.2iT - 7.92e3T^{2} \)
97 \( 1 - 22T + 9.40e3T^{2} \)
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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

−9.850593898804046294539661728391, −8.729308228141334049619387225024, −8.445714136487085271556949721215, −7.59903130782252000962467945800, −6.13352289307853883002477016007, −5.51654733592879906091759237707, −4.57271142789805412096121521407, −3.72404214635462295647506257984, −2.74543782384356575426931918215, −1.25478643712709726228180034542, 0.63629568886352041505168016092, 2.00170341477063596083510680171, 2.56127366772225805873811755341, 4.34345649447911130584429053018, 4.91749018845847083820293719435, 6.16244070696292573157067135268, 6.98276980673596953145490801971, 7.69802484254808182178168583935, 8.291703415065667236211511152864, 9.220392785961744204865643940803

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