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Properties

Label 2-600-24.11-c1-0-32
Degree $2$
Conductor $600$
Sign $0.562 - 0.826i$
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  + (−1.26 + 0.639i)2-s + (1.57 + 0.730i)3-s + (1.18 − 1.61i)4-s + (−2.44 + 0.0838i)6-s − 1.25i·7-s + (−0.458 + 2.79i)8-s + (1.93 + 2.29i)9-s − 3.02i·11-s + (3.03 − 1.67i)12-s + 5.65i·13-s + (0.803 + 1.58i)14-s + (−1.20 − 3.81i)16-s + 2.45i·17-s + (−3.90 − 1.65i)18-s + 1.77·19-s + ⋯
L(s)  = 1  + (−0.891 + 0.452i)2-s + (0.906 + 0.421i)3-s + (0.590 − 0.806i)4-s + (−0.999 + 0.0342i)6-s − 0.474i·7-s + (−0.162 + 0.986i)8-s + (0.644 + 0.764i)9-s − 0.911i·11-s + (0.875 − 0.482i)12-s + 1.56i·13-s + (0.214 + 0.423i)14-s + (−0.301 − 0.953i)16-s + 0.595i·17-s + (−0.920 − 0.390i)18-s + 0.407·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.562 - 0.826i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.562 - 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22940 + 0.650198i\)
\(L(\frac12)\) \(\approx\) \(1.22940 + 0.650198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 0.639i)T \)
3 \( 1 + (-1.57 - 0.730i)T \)
5 \( 1 \)
good7 \( 1 + 1.25iT - 7T^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 - 5.65iT - 13T^{2} \)
17 \( 1 - 2.45iT - 17T^{2} \)
19 \( 1 - 1.77T + 19T^{2} \)
23 \( 1 - 8.84T + 23T^{2} \)
29 \( 1 - 3.79T + 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 + 6.45iT - 37T^{2} \)
41 \( 1 + 7.57iT - 41T^{2} \)
43 \( 1 - 4.37T + 43T^{2} \)
47 \( 1 + 1.83T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 4.91iT - 59T^{2} \)
61 \( 1 - 8.16iT - 61T^{2} \)
67 \( 1 + 8.50T + 67T^{2} \)
71 \( 1 - 7.00T + 71T^{2} \)
73 \( 1 + 4.59T + 73T^{2} \)
79 \( 1 - 7.36iT - 79T^{2} \)
83 \( 1 - 15.7iT - 83T^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−10.76388803884973381829889797169, −9.628452369028765343096274093129, −8.977674666351312751938949824953, −8.418433987249476131660959547894, −7.31621396706233110910727281869, −6.69857139569296209386594116166, −5.34545552269549045395924448768, −4.15674258839741793705206751431, −2.86297690941946191305457273602, −1.42509976847702703621453727802, 1.12266409584375841745951544379, 2.60270675642621597050104604013, 3.20095950132259066413469768720, 4.79563033949099451749809273182, 6.36594288814599235565937853941, 7.40051820661082975543902653000, 7.918470548954518865218503980638, 8.865161207633759064579637682693, 9.565467481603088898762192134630, 10.26033547286019056299277244302

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