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Properties

Label 2-600-8.5-c1-0-27
Degree $2$
Conductor $600$
Sign $0.844 + 0.535i$
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  + (−0.264 + 1.38i)2-s + i·3-s + (−1.85 − 0.735i)4-s + (−1.38 − 0.264i)6-s − 0.941·7-s + (1.51 − 2.38i)8-s − 9-s − 4.49i·11-s + (0.735 − 1.85i)12-s − 5.55i·13-s + (0.249 − 1.30i)14-s + (2.91 + 2.73i)16-s − 7.55·17-s + (0.264 − 1.38i)18-s − 1.05i·19-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)2-s + 0.577i·3-s + (−0.929 − 0.367i)4-s + (−0.567 − 0.108i)6-s − 0.355·7-s + (0.535 − 0.844i)8-s − 0.333·9-s − 1.35i·11-s + (0.212 − 0.536i)12-s − 1.54i·13-s + (0.0665 − 0.349i)14-s + (0.729 + 0.683i)16-s − 1.83·17-s + (0.0623 − 0.327i)18-s − 0.242i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.844 + 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.651260 - 0.188950i\)
\(L(\frac12)\) \(\approx\) \(0.651260 - 0.188950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.264 - 1.38i)T \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 0.941T + 7T^{2} \)
11 \( 1 + 4.49iT - 11T^{2} \)
13 \( 1 + 5.55iT - 13T^{2} \)
17 \( 1 + 7.55T + 17T^{2} \)
19 \( 1 + 1.05iT - 19T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 3.55T + 31T^{2} \)
37 \( 1 + 7.43iT - 37T^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 + 1.88iT - 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 8.49iT - 59T^{2} \)
61 \( 1 + 8.99iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 5.88iT - 83T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−10.62196207829639554200986341326, −9.472634713628805229042339969457, −8.740828824538069222573806101469, −8.108855777583635825613059659581, −6.92376822750884238546988310007, −5.99771603995393831709951691866, −5.26757605239884486087795153915, −4.12792965582564463549581956595, −2.99840196858754915576763739664, −0.40122481678522826397575925102, 1.69092314152086572037839665490, 2.54372439705391079056865300081, 4.11422011800851363223938100233, 4.80050464605182267642646143433, 6.46560079821031382304761080687, 7.14398303748512547481259653013, 8.339659483327086462767242820680, 9.183765169426295738084510788782, 9.816960862770893040126799672234, 10.84670200317514538914236056616

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