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Properties

Label 2-600-25.4-c1-0-5
Degree $2$
Conductor $600$
Sign $0.988 - 0.149i$
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  + (−0.587 + 0.809i)3-s + (−2.23 − 0.113i)5-s − 1.02i·7-s + (−0.309 − 0.951i)9-s + (0.299 − 0.922i)11-s + (2.85 − 0.927i)13-s + (1.40 − 1.74i)15-s + (4.25 + 5.85i)17-s + (4.83 − 3.50i)19-s + (0.833 + 0.605i)21-s + (1.75 + 0.569i)23-s + (4.97 + 0.505i)25-s + (0.951 + 0.309i)27-s + (−0.296 − 0.215i)29-s + (−6.20 + 4.50i)31-s + ⋯
L(s)  = 1  + (−0.339 + 0.467i)3-s + (−0.998 − 0.0505i)5-s − 0.389i·7-s + (−0.103 − 0.317i)9-s + (0.0903 − 0.278i)11-s + (0.791 − 0.257i)13-s + (0.362 − 0.449i)15-s + (1.03 + 1.42i)17-s + (1.10 − 0.805i)19-s + (0.181 + 0.132i)21-s + (0.365 + 0.118i)23-s + (0.994 + 0.101i)25-s + (0.183 + 0.0594i)27-s + (−0.0549 − 0.0399i)29-s + (−1.11 + 0.809i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.988 - 0.149i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15103 + 0.0865629i\)
\(L(\frac12)\) \(\approx\) \(1.15103 + 0.0865629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (2.23 + 0.113i)T \)
good7 \( 1 + 1.02iT - 7T^{2} \)
11 \( 1 + (-0.299 + 0.922i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-2.85 + 0.927i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.25 - 5.85i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.83 + 3.50i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.75 - 0.569i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.296 + 0.215i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (6.20 - 4.50i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-9.38 + 3.05i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.12 - 6.53i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.23iT - 43T^{2} \)
47 \( 1 + (-1.44 + 1.98i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.734 - 1.01i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.49 + 13.8i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.30 + 4.02i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.86 - 3.94i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (0.549 + 0.399i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-5.85 - 1.90i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (10.2 + 7.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.35 - 11.4i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.47 + 4.55i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.00 - 6.88i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.97132875428546313006466755898, −9.911845138108645609766655643694, −8.904317252265834322453092973928, −8.035914095593415878532769941385, −7.22499716245368315112844900506, −6.06416983251304266360539537314, −5.09244111561420047861392060701, −3.91669937946222209905385472595, −3.29585715743792302392530887423, −0.979463258555049556344286126948, 1.03487713069178903828934358326, 2.83737635662119842042257563985, 3.97004844683650894725676606070, 5.19109480581991492566634193315, 6.08645293367469075761843630440, 7.37277130302052248826080229638, 7.67479978382821173161608049712, 8.863590732879075218559834942861, 9.714051062533121682250907518417, 10.90524802273109614098085269919

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