Properties

Label 2-600-40.29-c1-0-26
Degree $2$
Conductor $600$
Sign $0.852 + 0.522i$
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  + (1.24 − 0.671i)2-s + 3-s + (1.09 − 1.67i)4-s + (1.24 − 0.671i)6-s + 4.68i·7-s + (0.244 − 2.81i)8-s + 9-s − 2.29i·11-s + (1.09 − 1.67i)12-s + 4.97·13-s + (3.14 + 5.83i)14-s + (−1.58 − 3.67i)16-s − 2.97i·17-s + (1.24 − 0.671i)18-s + 2.68i·19-s + ⋯
L(s)  = 1  + (0.880 − 0.474i)2-s + 0.577·3-s + (0.549 − 0.835i)4-s + (0.508 − 0.274i)6-s + 1.77i·7-s + (0.0864 − 0.996i)8-s + 0.333·9-s − 0.691i·11-s + (0.317 − 0.482i)12-s + 1.38·13-s + (0.840 + 1.55i)14-s + (−0.396 − 0.917i)16-s − 0.722i·17-s + (0.293 − 0.158i)18-s + 0.616i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91328 - 0.822374i\)
\(L(\frac12)\) \(\approx\) \(2.91328 - 0.822374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 + 0.671i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 - 4.68iT - 7T^{2} \)
11 \( 1 + 2.29iT - 11T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 + 2.97iT - 17T^{2} \)
19 \( 1 - 2.68iT - 19T^{2} \)
23 \( 1 + 2.68iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 - 4.39T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 9.37T + 43T^{2} \)
47 \( 1 - 7.27iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 1.70iT - 59T^{2} \)
61 \( 1 + 4.58iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 0.585T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 1.02T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 + 3.95iT - 97T^{2} \)
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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

−10.84955405386415126659492257479, −9.684044360727980832230064250618, −8.879883947830220902457173992825, −8.189595967031247087359708588524, −6.65554910077042468251844151102, −5.85474671745086958442085103907, −5.08069673183854232950890688261, −3.64469976760805615710973763927, −2.86510314507904884962965152604, −1.70706145026574424906716649019, 1.69841650150167190005975056670, 3.49765089720432791333809877007, 3.94742907484286548935946397024, 5.01823876768522536197359845525, 6.41273348984488609322165730348, 7.10185490142539487624984042017, 7.86506811036578009990644802577, 8.715472323445342519948238649533, 10.01370651112997052809573112429, 10.80862653420548213591625336159

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