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Properties

Label 2-1200-60.59-c1-0-10
Degree $2$
Conductor $1200$
Sign $0.894 - 0.447i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 1.5i)3-s + (−1.5 + 2.59i)9-s − 3·11-s + 2i·13-s + 5.19·17-s + 5.19i·19-s − 6i·23-s + 5.19·27-s + 10.3i·29-s + 3.46i·31-s + (2.59 + 4.5i)33-s + 8i·37-s + (3 − 1.73i)39-s − 5.19i·41-s + 3.46·43-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (−0.5 + 0.866i)9-s − 0.904·11-s + 0.554i·13-s + 1.26·17-s + 1.19i·19-s − 1.25i·23-s + 1.00·27-s + 1.92i·29-s + 0.622i·31-s + (0.452 + 0.783i)33-s + 1.31i·37-s + (0.480 − 0.277i)39-s − 0.811i·41-s + 0.528·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.095209602\)
\(L(\frac12)\) \(\approx\) \(1.095209602\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 5.19T + 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 - 6.92iT - 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 - 5.19iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.14498060496695593370527366224, −8.711012657288300815383131350536, −8.142074131355289712251270998573, −7.25916532236518643204142462270, −6.57915888707377329483519794594, −5.55969291465058352616160737099, −4.98534942806625248997553058568, −3.53578843283938185265770384955, −2.35813636731641701614943154646, −1.15855995616410856790063049314, 0.57839160716845157709482419622, 2.58563794369830486514893799642, 3.57142530586338997236281218908, 4.59021427995216149536628631711, 5.50799559078641462478149683067, 5.97912177988653766011799311559, 7.34219746356683942611023931540, 8.002634532752894884968670307168, 9.085105335909384246348012378107, 9.860371660842822433137428629175

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