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Properties

Label 2-1200-5.3-c2-0-9
Degree $2$
Conductor $1200$
Sign $0.229 - 0.973i$
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 + (−2.57 + 2.57i)7-s + 2.99i·9-s + 21.2·11-s + (8.47 + 8.47i)13-s + (−19.3 + 19.3i)17-s − 13.1i·19-s + 6.30·21-s + (−19.2 − 19.2i)23-s + (3.67 − 3.67i)27-s − 1.00i·29-s − 23.3·31-s + (−26.0 − 26.0i)33-s + (−29.8 + 29.8i)37-s − 20.7i·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.367 + 0.367i)7-s + 0.333i·9-s + 1.93·11-s + (0.651 + 0.651i)13-s + (−1.13 + 1.13i)17-s − 0.692i·19-s + 0.300·21-s + (−0.834 − 0.834i)23-s + (0.136 − 0.136i)27-s − 0.0346i·29-s − 0.751·31-s + (−0.788 − 0.788i)33-s + (−0.807 + 0.807i)37-s − 0.532i·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.292492589\)
\(L(\frac12)\) \(\approx\) \(1.292492589\)
\(L(2)\) 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 \( 1 + (2.57 - 2.57i)T - 49iT^{2} \)
11 \( 1 - 21.2T + 121T^{2} \)
13 \( 1 + (-8.47 - 8.47i)T + 169iT^{2} \)
17 \( 1 + (19.3 - 19.3i)T - 289iT^{2} \)
19 \( 1 + 13.1iT - 361T^{2} \)
23 \( 1 + (19.2 + 19.2i)T + 529iT^{2} \)
29 \( 1 + 1.00iT - 841T^{2} \)
31 \( 1 + 23.3T + 961T^{2} \)
37 \( 1 + (29.8 - 29.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 0.555T + 1.68e3T^{2} \)
43 \( 1 + (-8.85 - 8.85i)T + 1.84e3iT^{2} \)
47 \( 1 + (-9.53 + 9.53i)T - 2.20e3iT^{2} \)
53 \( 1 + (-34.4 - 34.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 47.0iT - 3.48e3T^{2} \)
61 \( 1 + 64.3T + 3.72e3T^{2} \)
67 \( 1 + (-12.4 + 12.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 91.2T + 5.04e3T^{2} \)
73 \( 1 + (-77.6 - 77.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 + (-50.0 - 50.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 163. iT - 7.92e3T^{2} \)
97 \( 1 + (-100. + 100. i)T - 9.40e3iT^{2} \)
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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

−9.535869971741751710061383064672, −8.921599336084506006644992320312, −8.277007667792564585351499518541, −6.79666555906911480368695355421, −6.58502849953393460905604386268, −5.79236290492980193821381647611, −4.39862067165967692585320004107, −3.78161436355158805027030905304, −2.23027452526585157698022246993, −1.22270450982872365102780179348, 0.44166232721552447955394099363, 1.78434199304439915270339616005, 3.52605076462191525483314348681, 3.94776489612314495957236238939, 5.12310473239863186297223155282, 6.12739385389375326461975004792, 6.72445555946677823286562257824, 7.64054674304441991487875417983, 8.893655458898626471192049475578, 9.306088396378944723631251229064

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