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Properties

Label 2-1200-5.2-c2-0-27
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 + (2.55 + 2.55i)7-s − 2.99i·9-s − 8.24·11-s + (12.2 − 12.2i)13-s + (12.4 + 12.4i)17-s − 34.4i·19-s + 6.24·21-s + (−17.3 + 17.3i)23-s + (−3.67 − 3.67i)27-s − 9.75i·29-s − 28.4·31-s + (−10.1 + 10.1i)33-s + (7.34 + 7.34i)37-s − 29.9i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.364 + 0.364i)7-s − 0.333i·9-s − 0.749·11-s + (0.942 − 0.942i)13-s + (0.732 + 0.732i)17-s − 1.81i·19-s + 0.297·21-s + (−0.754 + 0.754i)23-s + (−0.136 − 0.136i)27-s − 0.336i·29-s − 0.919·31-s + (−0.306 + 0.306i)33-s + (0.198 + 0.198i)37-s − 0.769i·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: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.154095412\)
\(L(\frac12)\) \(\approx\) \(2.154095412\)
\(L(2)\) 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 \( 1 + (-2.55 - 2.55i)T + 49iT^{2} \)
11 \( 1 + 8.24T + 121T^{2} \)
13 \( 1 + (-12.2 + 12.2i)T - 169iT^{2} \)
17 \( 1 + (-12.4 - 12.4i)T + 289iT^{2} \)
19 \( 1 + 34.4iT - 361T^{2} \)
23 \( 1 + (17.3 - 17.3i)T - 529iT^{2} \)
29 \( 1 + 9.75iT - 841T^{2} \)
31 \( 1 + 28.4T + 961T^{2} \)
37 \( 1 + (-7.34 - 7.34i)T + 1.36e3iT^{2} \)
41 \( 1 - 74.4T + 1.68e3T^{2} \)
43 \( 1 + (-34.8 + 34.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (22.0 + 22.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-64.6 + 64.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 15.2iT - 3.48e3T^{2} \)
61 \( 1 + 53.5T + 3.72e3T^{2} \)
67 \( 1 + (-4.69 - 4.69i)T + 4.48e3iT^{2} \)
71 \( 1 - 117.T + 5.04e3T^{2} \)
73 \( 1 + (34.1 - 34.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 0.494iT - 6.24e3T^{2} \)
83 \( 1 + (18.3 - 18.3i)T - 6.88e3iT^{2} \)
89 \( 1 + 136. iT - 7.92e3T^{2} \)
97 \( 1 + (94.5 + 94.5i)T + 9.40e3iT^{2} \)
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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

−9.238383447742842815826268748756, −8.445091140583236075835538056170, −7.85988322691184495028576389774, −7.06799467242259103279023226597, −5.87485542425497325434031468932, −5.34625930666839855881951610136, −4.00868297928116944703372235013, −3.01052618829453569147523090559, −2.01609492896569505111277336682, −0.64970099689296551964658317350, 1.27346393831714704793864979376, 2.50340015227475133104046525077, 3.72249922097480972227700420952, 4.36460380462326865425964409903, 5.50941904657791516252109867042, 6.28645157402921710735504303515, 7.63510692532436493660923526495, 7.930036240751853034383194139702, 9.003070267208789304720519053854, 9.664122712615453809237143386791

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