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Properties

Label 2-1200-5.2-c2-0-32
Degree $2$
Conductor $1200$
Sign $-0.973 + 0.229i$
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  + (1.22 − 1.22i)3-s + (1.55 + 1.55i)7-s − 2.99i·9-s − 7.79·11-s + (−2.44 + 2.44i)13-s + (−8.89 − 8.89i)17-s + 5.59i·19-s + 3.79·21-s + (2.20 − 2.20i)23-s + (−3.67 − 3.67i)27-s − 35.5i·29-s − 53.1·31-s + (−9.55 + 9.55i)33-s + (−25.1 − 25.1i)37-s + 5.99i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.221 + 0.221i)7-s − 0.333i·9-s − 0.708·11-s + (−0.188 + 0.188i)13-s + (−0.523 − 0.523i)17-s + 0.294i·19-s + 0.180·21-s + (0.0957 − 0.0957i)23-s + (−0.136 − 0.136i)27-s − 1.22i·29-s − 1.71·31-s + (−0.289 + 0.289i)33-s + (−0.679 − 0.679i)37-s + 0.153i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6222305667\)
\(L(\frac12)\) \(\approx\) \(0.6222305667\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-1.55 - 1.55i)T + 49iT^{2} \)
11 \( 1 + 7.79T + 121T^{2} \)
13 \( 1 + (2.44 - 2.44i)T - 169iT^{2} \)
17 \( 1 + (8.89 + 8.89i)T + 289iT^{2} \)
19 \( 1 - 5.59iT - 361T^{2} \)
23 \( 1 + (-2.20 + 2.20i)T - 529iT^{2} \)
29 \( 1 + 35.5iT - 841T^{2} \)
31 \( 1 + 53.1T + 961T^{2} \)
37 \( 1 + (25.1 + 25.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 56.7T + 1.68e3T^{2} \)
43 \( 1 + (41.7 - 41.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (44.4 + 44.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (36.0 - 36.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 10.9iT - 3.48e3T^{2} \)
61 \( 1 - 48.3T + 3.72e3T^{2} \)
67 \( 1 + (29.8 + 29.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 50.7T + 5.04e3T^{2} \)
73 \( 1 + (-49.3 + 49.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 66iT - 6.24e3T^{2} \)
83 \( 1 + (4.09 - 4.09i)T - 6.88e3iT^{2} \)
89 \( 1 + 29.5iT - 7.92e3T^{2} \)
97 \( 1 + (133. + 133. i)T + 9.40e3iT^{2} \)
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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

−9.151512688614433500162337556857, −8.291534567496280491325395769260, −7.59924067609983717627210798356, −6.82368066432541184947905035629, −5.80132241049944495454685091268, −4.92738460129293001445860707059, −3.82461859843883811707593323968, −2.67661399124335091995316909737, −1.80102113429668621206935876758, −0.15984864011543995600087997804, 1.65011986435026234548087957195, 2.83573061144667286643712788808, 3.78978442847130694811631448167, 4.81890259208299077785154254331, 5.52986456028439871533524391148, 6.74850402592090170773691277156, 7.56486388288319902893322382070, 8.361162329630199939305607849555, 9.098161785775858816086733987831, 9.909112650895434730147188378452

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