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Properties

Label 2-600-25.9-c1-0-8
Degree $2$
Conductor $600$
Sign $0.994 + 0.104i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)3-s + (1.33 + 1.79i)5-s − 2.61i·7-s + (0.809 − 0.587i)9-s + (0.724 + 0.526i)11-s + (−0.0463 − 0.0638i)13-s + (1.82 + 1.29i)15-s + (5.69 + 1.84i)17-s + (1.20 − 3.69i)19-s + (−0.808 − 2.48i)21-s + (1.49 − 2.05i)23-s + (−1.44 + 4.78i)25-s + (0.587 − 0.809i)27-s + (−0.579 − 1.78i)29-s + (−1.91 + 5.90i)31-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s + (0.596 + 0.802i)5-s − 0.988i·7-s + (0.269 − 0.195i)9-s + (0.218 + 0.158i)11-s + (−0.0128 − 0.0176i)13-s + (0.470 + 0.334i)15-s + (1.38 + 0.448i)17-s + (0.275 − 0.847i)19-s + (−0.176 − 0.542i)21-s + (0.311 − 0.429i)23-s + (−0.288 + 0.957i)25-s + (0.113 − 0.155i)27-s + (−0.107 − 0.330i)29-s + (−0.344 + 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.994 + 0.104i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.994 + 0.104i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02493 - 0.105700i\)
\(L(\frac12)\) \(\approx\) \(2.02493 - 0.105700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (-1.33 - 1.79i)T \)
good7 \( 1 + 2.61iT - 7T^{2} \)
11 \( 1 + (-0.724 - 0.526i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.0463 + 0.0638i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.69 - 1.84i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.20 + 3.69i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.49 + 2.05i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.579 + 1.78i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.91 - 5.90i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.201 - 0.277i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.83 - 5.69i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.79iT - 43T^{2} \)
47 \( 1 + (-4.21 + 1.36i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.47 + 0.480i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.57 + 6.22i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.78 + 6.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.75 + 1.21i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.22 + 9.92i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.92 - 8.16i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.15 + 3.56i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (6.63 + 2.15i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-3.32 - 2.41i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.13 + 0.695i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.42970727838563795024142474655, −9.945089126609579843234113614072, −8.980089240522453711273491742437, −7.83736905823102741294740248486, −7.11350805355322019580058269539, −6.36740433866530888166917857071, −5.09451327360808219852196149595, −3.75247500675040677310411310140, −2.87289416960837460236938047575, −1.40470135450293862021414419198, 1.49471509909424278773909630086, 2.76705646354898624632714771357, 3.99821353289162458031588342424, 5.39920809821517255130696453724, 5.75065410356757723439608951325, 7.27433574534945734624816844533, 8.264983931881148523698633788436, 8.995738415374952258607480903997, 9.626903409149234008348091219266, 10.40665023068279164204613124162

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