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Properties

Label 2-600-5.4-c3-0-18
Degree $2$
Conductor $600$
Sign $0.447 + 0.894i$
Analytic cond. $35.4011$
Root an. cond. $5.94988$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 20i·7-s − 9·9-s + 16·11-s + 58i·13-s − 38i·17-s − 4·19-s + 60·21-s − 80i·23-s − 27i·27-s − 82·29-s − 8·31-s + 48i·33-s − 426i·37-s − 174·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.07i·7-s − 0.333·9-s + 0.438·11-s + 1.23i·13-s − 0.542i·17-s − 0.0482·19-s + 0.623·21-s − 0.725i·23-s − 0.192i·27-s − 0.525·29-s − 0.0463·31-s + 0.253i·33-s − 1.89i·37-s − 0.714·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(35.4011\)
Root analytic conductor: \(5.94988\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.517079692\)
\(L(\frac12)\) \(\approx\) \(1.517079692\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 + 20iT - 343T^{2} \)
11 \( 1 - 16T + 1.33e3T^{2} \)
13 \( 1 - 58iT - 2.19e3T^{2} \)
17 \( 1 + 38iT - 4.91e3T^{2} \)
19 \( 1 + 4T + 6.85e3T^{2} \)
23 \( 1 + 80iT - 1.21e4T^{2} \)
29 \( 1 + 82T + 2.43e4T^{2} \)
31 \( 1 + 8T + 2.97e4T^{2} \)
37 \( 1 + 426iT - 5.06e4T^{2} \)
41 \( 1 + 246T + 6.89e4T^{2} \)
43 \( 1 + 524iT - 7.95e4T^{2} \)
47 \( 1 - 464iT - 1.03e5T^{2} \)
53 \( 1 + 702iT - 1.48e5T^{2} \)
59 \( 1 - 592T + 2.05e5T^{2} \)
61 \( 1 - 574T + 2.26e5T^{2} \)
67 \( 1 - 172iT - 3.00e5T^{2} \)
71 \( 1 - 768T + 3.57e5T^{2} \)
73 \( 1 + 558iT - 3.89e5T^{2} \)
79 \( 1 + 408T + 4.93e5T^{2} \)
83 \( 1 - 164iT - 5.71e5T^{2} \)
89 \( 1 - 510T + 7.04e5T^{2} \)
97 \( 1 + 514iT - 9.12e5T^{2} \)
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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

−10.08925464537923838124710758917, −9.323707853672958177011675162978, −8.519696114425871390472065603652, −7.26368244895566834683292449706, −6.67712189287014130888119718729, −5.36659441327925108958476994143, −4.27404490811162402601632327544, −3.68203004603728928782240578666, −2.07097017686324727908071753403, −0.47834145687975776989485504512, 1.21863089023482092184466507891, 2.48838017547789598607606052442, 3.54358853631334985851693434924, 5.11891543963230836061380993778, 5.87403071882933770742035925329, 6.75457906899322039530647368671, 7.929228793889385178004885778900, 8.501449951883615070174018474750, 9.481481918143408538412420897794, 10.39440214595026171084965836447

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