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Properties

Label 2-1200-5.2-c2-0-30
Degree $2$
Conductor $1200$
Sign $-0.326 + 0.945i$
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 + (−3.22 − 3.22i)7-s − 2.99i·9-s + 6.89·11-s + (18.1 − 18.1i)13-s + (−0.449 − 0.449i)17-s − 9.89i·19-s − 7.89·21-s + (−10.6 + 10.6i)23-s + (−3.67 − 3.67i)27-s + 36.2i·29-s − 25.6·31-s + (8.44 − 8.44i)33-s + (13.3 + 13.3i)37-s − 44.3i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.460 − 0.460i)7-s − 0.333i·9-s + 0.627·11-s + (1.39 − 1.39i)13-s + (−0.0264 − 0.0264i)17-s − 0.520i·19-s − 0.376·21-s + (−0.463 + 0.463i)23-s + (−0.136 − 0.136i)27-s + 1.25i·29-s − 0.828·31-s + (0.256 − 0.256i)33-s + (0.359 + 0.359i)37-s − 1.13i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.326 + 0.945i$
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: \((2,\ 1200,\ (\ :1),\ -0.326 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.975408545\)
\(L(\frac12)\) \(\approx\) \(1.975408545\)
\(L(2)\) 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 \( 1 + (3.22 + 3.22i)T + 49iT^{2} \)
11 \( 1 - 6.89T + 121T^{2} \)
13 \( 1 + (-18.1 + 18.1i)T - 169iT^{2} \)
17 \( 1 + (0.449 + 0.449i)T + 289iT^{2} \)
19 \( 1 + 9.89iT - 361T^{2} \)
23 \( 1 + (10.6 - 10.6i)T - 529iT^{2} \)
29 \( 1 - 36.2iT - 841T^{2} \)
31 \( 1 + 25.6T + 961T^{2} \)
37 \( 1 + (-13.3 - 13.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 3.10T + 1.68e3T^{2} \)
43 \( 1 + (2.72 - 2.72i)T - 1.84e3iT^{2} \)
47 \( 1 + (37.1 + 37.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (-65.1 + 65.1i)T - 2.80e3iT^{2} \)
59 \( 1 + 80.3iT - 3.48e3T^{2} \)
61 \( 1 - 13.7T + 3.72e3T^{2} \)
67 \( 1 + (84.3 + 84.3i)T + 4.48e3iT^{2} \)
71 \( 1 + 98.2T + 5.04e3T^{2} \)
73 \( 1 + (-52.4 + 52.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 68.2iT - 6.24e3T^{2} \)
83 \( 1 + (-89.7 + 89.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 40.5iT - 7.92e3T^{2} \)
97 \( 1 + (-105. - 105. i)T + 9.40e3iT^{2} \)
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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

−9.175950220189939499782876814612, −8.500843895067561834304805474641, −7.70266743411458723591577926271, −6.82007629651085567148950229288, −6.11485160325057765988790158427, −5.10723166357050988289221628564, −3.66558367665814628433992545070, −3.27600747922711905009195735092, −1.72175750309240181507364025904, −0.57429282979663121938557124706, 1.44491148853257613938960752596, 2.60914190729332968386700622030, 3.87001633526596544159031368273, 4.28077250410374239958765724571, 5.81387707754950799651409222948, 6.30914226145723935735824519479, 7.35399711316215276931635960546, 8.447416251091301310262374864213, 9.008397790925584555187321022542, 9.612476684701573385801533005609

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