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Properties

Label 2-600-15.2-c1-0-6
Degree $2$
Conductor $600$
Sign $0.975 - 0.220i$
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  + (−0.912 − 1.47i)3-s + (1.78 + 1.78i)7-s + (−1.33 + 2.68i)9-s + 4.25i·11-s + (2.90 − 2.90i)13-s + (−0.443 + 0.443i)17-s + 7.74i·19-s + (1 − 4.25i)21-s + (−1.94 − 1.94i)23-s + (5.17 − 0.483i)27-s + 10.0·29-s + 0.372·31-s + (6.26 − 3.88i)33-s + (1.12 + 1.12i)37-s + (−6.92 − 1.62i)39-s + ⋯
L(s)  = 1  + (−0.526 − 0.850i)3-s + (0.674 + 0.674i)7-s + (−0.445 + 0.895i)9-s + 1.28i·11-s + (0.805 − 0.805i)13-s + (−0.107 + 0.107i)17-s + 1.77i·19-s + (0.218 − 0.928i)21-s + (−0.404 − 0.404i)23-s + (0.995 − 0.0929i)27-s + 1.87·29-s + 0.0668·31-s + (1.09 − 0.675i)33-s + (0.184 + 0.184i)37-s + (−1.10 − 0.260i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.975 - 0.220i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29022 + 0.143897i\)
\(L(\frac12)\) \(\approx\) \(1.29022 + 0.143897i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.912 + 1.47i)T \)
5 \( 1 \)
good7 \( 1 + (-1.78 - 1.78i)T + 7iT^{2} \)
11 \( 1 - 4.25iT - 11T^{2} \)
13 \( 1 + (-2.90 + 2.90i)T - 13iT^{2} \)
17 \( 1 + (0.443 - 0.443i)T - 17iT^{2} \)
19 \( 1 - 7.74iT - 19T^{2} \)
23 \( 1 + (1.94 + 1.94i)T + 23iT^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 - 0.372T + 31T^{2} \)
37 \( 1 + (-1.12 - 1.12i)T + 37iT^{2} \)
41 \( 1 + 7.42iT - 41T^{2} \)
43 \( 1 + (-6.68 + 6.68i)T - 43iT^{2} \)
47 \( 1 + (5.65 - 5.65i)T - 47iT^{2} \)
53 \( 1 + (-7.59 - 7.59i)T + 53iT^{2} \)
59 \( 1 - 5.34T + 59T^{2} \)
61 \( 1 + 4.37T + 61T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (7.90 - 7.90i)T - 73iT^{2} \)
79 \( 1 + 2.74iT - 79T^{2} \)
83 \( 1 + (8.04 + 8.04i)T + 83iT^{2} \)
89 \( 1 + 0.497T + 89T^{2} \)
97 \( 1 + (6.47 + 6.47i)T + 97iT^{2} \)
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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

−10.65388885458228607509324499553, −10.10749491320197102317518237053, −8.605801712602226040726077397287, −8.076730297060735042018522904830, −7.15032733444278787882336604655, −6.07392989240676584549689729441, −5.40377058273549346928447345559, −4.24018907600837719732370411015, −2.50306068087129602774623824703, −1.40387282466597563740800407484, 0.902007789345695705796302782921, 3.02613241672187577577561065981, 4.19058341775755399838370979219, 4.90001681721503020828747113773, 6.07675461480089933329778747911, 6.83802788886700742221490314610, 8.233799181453032973856719538234, 8.900931070098198000375441417607, 9.842341793194400158050155038261, 10.93262822380004031536048509883

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