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Properties

Label 2-1200-3.2-c2-0-63
Degree $2$
Conductor $1200$
Sign $-0.800 - 0.599i$
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  + (−2.40 − 1.79i)3-s − 10.2·7-s + (2.53 + 8.63i)9-s − 8.19i·11-s + 13.5·13-s − 15.4i·17-s + 25.4·19-s + (24.5 + 18.3i)21-s − 17.9i·23-s + (9.44 − 25.2i)27-s − 42.0i·29-s − 38.4·31-s + (−14.7 + 19.6i)33-s − 11.8·37-s + (−32.6 − 24.4i)39-s + ⋯
L(s)  = 1  + (−0.800 − 0.599i)3-s − 1.45·7-s + (0.281 + 0.959i)9-s − 0.744i·11-s + 1.04·13-s − 0.910i·17-s + 1.34·19-s + (1.16 + 0.874i)21-s − 0.778i·23-s + (0.349 − 0.936i)27-s − 1.44i·29-s − 1.24·31-s + (−0.446 + 0.596i)33-s − 0.319·37-s + (−0.836 − 0.626i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.800 - 0.599i$
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: \((2,\ 1200,\ (\ :1),\ -0.800 - 0.599i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1721025671\)
\(L(\frac12)\) \(\approx\) \(0.1721025671\)
\(L(2)\) 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 \( 1 + (2.40 + 1.79i)T \)
5 \( 1 \)
good7 \( 1 + 10.2T + 49T^{2} \)
11 \( 1 + 8.19iT - 121T^{2} \)
13 \( 1 - 13.5T + 169T^{2} \)
17 \( 1 + 15.4iT - 289T^{2} \)
19 \( 1 - 25.4T + 361T^{2} \)
23 \( 1 + 17.9iT - 529T^{2} \)
29 \( 1 + 42.0iT - 841T^{2} \)
31 \( 1 + 38.4T + 961T^{2} \)
37 \( 1 + 11.8T + 1.36e3T^{2} \)
41 \( 1 - 46.3iT - 1.68e3T^{2} \)
43 \( 1 + 54.0T + 1.84e3T^{2} \)
47 \( 1 - 43.0iT - 2.20e3T^{2} \)
53 \( 1 - 82.7iT - 2.80e3T^{2} \)
59 \( 1 + 45.8iT - 3.48e3T^{2} \)
61 \( 1 + 93.6T + 3.72e3T^{2} \)
67 \( 1 - 34.4T + 4.48e3T^{2} \)
71 \( 1 + 68.0iT - 5.04e3T^{2} \)
73 \( 1 - 44.7T + 5.32e3T^{2} \)
79 \( 1 - 11.7T + 6.24e3T^{2} \)
83 \( 1 - 144. iT - 6.88e3T^{2} \)
89 \( 1 + 63.7iT - 7.92e3T^{2} \)
97 \( 1 + 63.9T + 9.40e3T^{2} \)
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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

−9.220403136861160119284425444568, −8.107475734071758365022937668854, −7.27454301939710255463667915100, −6.34393606713949032061025004338, −5.99808958232003056730707318488, −4.95952950524627156084210447080, −3.62525903305469021588687420467, −2.74582469023170673780311208237, −1.10921646335036839843212271484, −0.06696985084232075874188191739, 1.48270256334560020027324843798, 3.42938618515720240728178950562, 3.69229933525715734161955668734, 5.13426615135959071737680543608, 5.78820972280147719404323882299, 6.68166382837162688638810285798, 7.26952712120480157486397553924, 8.715337609581046470743335415776, 9.400681993091180685377028983860, 10.06832946756910525504318090457

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