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Properties

Label 2-600-40.29-c1-0-23
Degree $2$
Conductor $600$
Sign $0.943 - 0.331i$
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  + (1.41 + 0.0591i)2-s + 3-s + (1.99 + 0.167i)4-s + (1.41 + 0.0591i)6-s + 1.33i·7-s + (2.80 + 0.353i)8-s + 9-s + 2.94i·11-s + (1.99 + 0.167i)12-s − 2.04·13-s + (−0.0788 + 1.88i)14-s + (3.94 + 0.665i)16-s + 3.61i·17-s + (1.41 + 0.0591i)18-s − 5.35i·19-s + ⋯
L(s)  = 1  + (0.999 + 0.0418i)2-s + 0.577·3-s + (0.996 + 0.0835i)4-s + (0.576 + 0.0241i)6-s + 0.504i·7-s + (0.992 + 0.125i)8-s + 0.333·9-s + 0.887i·11-s + (0.575 + 0.0482i)12-s − 0.566·13-s + (−0.0210 + 0.503i)14-s + (0.986 + 0.166i)16-s + 0.876i·17-s + (0.333 + 0.0139i)18-s − 1.22i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.943 - 0.331i$
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: \((2,\ 600,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.22440 + 0.550519i\)
\(L(\frac12)\) \(\approx\) \(3.22440 + 0.550519i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0591i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 - 1.33iT - 7T^{2} \)
11 \( 1 - 2.94iT - 11T^{2} \)
13 \( 1 + 2.04T + 13T^{2} \)
17 \( 1 - 3.61iT - 17T^{2} \)
19 \( 1 + 5.35iT - 19T^{2} \)
23 \( 1 + 8.59iT - 23T^{2} \)
29 \( 1 + 5.26iT - 29T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 + 6.55T + 37T^{2} \)
41 \( 1 - 7.02T + 41T^{2} \)
43 \( 1 + 8.50T + 43T^{2} \)
47 \( 1 - 9.97iT - 47T^{2} \)
53 \( 1 + 6.12T + 53T^{2} \)
59 \( 1 + 4.75iT - 59T^{2} \)
61 \( 1 + 8.51iT - 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 2.62T + 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 - 1.52T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−10.78193691255046807763068438041, −9.959243276377854374841850050073, −8.899421555619399293029814345359, −7.922339506644431173764879170292, −7.00615887324707667942640613326, −6.18459397964407310273768024465, −4.91841396472312172419833378779, −4.23634401932990061232852529177, −2.86085569548846092714062913626, −2.03310905206904008155136504438, 1.62156460732819543500761250582, 3.10983942703431154316713992371, 3.76862831431111336616735592613, 5.02809339250815931590734770693, 5.88175235268422479899289774330, 7.15002834793942196828727274863, 7.63962422382198120006857997793, 8.835110364653254064812909049438, 9.936092604230651304639103313263, 10.68282207143527938738732743944

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