Properties

Label 2-600-3.2-c2-0-10
Degree $2$
Conductor $600$
Sign $0.312 - 0.949i$
Analytic cond. $16.3488$
Root an. cond. $4.04336$
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  + (−2.84 − 0.938i)3-s + 6.81·7-s + (7.23 + 5.34i)9-s + 7.52i·11-s − 16.2·13-s − 4.11i·17-s − 7.86·19-s + (−19.4 − 6.39i)21-s − 19.5i·23-s + (−15.6 − 22.0i)27-s + 55.8i·29-s + 43.4·31-s + (7.06 − 21.4i)33-s − 31.5·37-s + (46.2 + 15.2i)39-s + ⋯
L(s)  = 1  + (−0.949 − 0.312i)3-s + 0.973·7-s + (0.804 + 0.594i)9-s + 0.684i·11-s − 1.24·13-s − 0.242i·17-s − 0.413·19-s + (−0.924 − 0.304i)21-s − 0.848i·23-s + (−0.578 − 0.815i)27-s + 1.92i·29-s + 1.40·31-s + (0.214 − 0.650i)33-s − 0.853·37-s + (1.18 + 0.390i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.312 - 0.949i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.312 - 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.034952638\)
\(L(\frac12)\) \(\approx\) \(1.034952638\)
\(L(2)\) 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 \( 1 + (2.84 + 0.938i)T \)
5 \( 1 \)
good7 \( 1 - 6.81T + 49T^{2} \)
11 \( 1 - 7.52iT - 121T^{2} \)
13 \( 1 + 16.2T + 169T^{2} \)
17 \( 1 + 4.11iT - 289T^{2} \)
19 \( 1 + 7.86T + 361T^{2} \)
23 \( 1 + 19.5iT - 529T^{2} \)
29 \( 1 - 55.8iT - 841T^{2} \)
31 \( 1 - 43.4T + 961T^{2} \)
37 \( 1 + 31.5T + 1.36e3T^{2} \)
41 \( 1 - 51.3iT - 1.68e3T^{2} \)
43 \( 1 - 51.2T + 1.84e3T^{2} \)
47 \( 1 - 61.7iT - 2.20e3T^{2} \)
53 \( 1 - 82.7iT - 2.80e3T^{2} \)
59 \( 1 - 97.6iT - 3.48e3T^{2} \)
61 \( 1 - 4.13T + 3.72e3T^{2} \)
67 \( 1 - 63.1T + 4.48e3T^{2} \)
71 \( 1 + 40.3iT - 5.04e3T^{2} \)
73 \( 1 + 78.5T + 5.32e3T^{2} \)
79 \( 1 - 51.0T + 6.24e3T^{2} \)
83 \( 1 - 2.72iT - 6.88e3T^{2} \)
89 \( 1 + 70.4iT - 7.92e3T^{2} \)
97 \( 1 - 3.44T + 9.40e3T^{2} \)
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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

−10.67029720228699956048231709747, −10.03470085641950757274650762842, −8.866809768137054366881429753362, −7.72040857444457193324481445056, −7.12590019018097502074422621056, −6.11015188571404323311469711827, −4.84043954642962917070198591788, −4.60860533745454495109270812438, −2.52829389847606497412828907473, −1.26328109814968183858291089520, 0.48007488244534129650297865342, 2.09919146711306146099553454641, 3.82703929744634877174261275878, 4.83120313699214708430639410917, 5.55135164987865776634010694075, 6.57165129149733740058200324299, 7.60224068921220156314334019981, 8.471000890311816504454671785057, 9.678286324866104341303565551785, 10.31276717794054900638611904004

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