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Properties

Label 2-600-600.437-c1-0-51
Degree $2$
Conductor $600$
Sign $0.647 + 0.762i$
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  + (0.221 + 1.39i)2-s + (−1.54 − 0.786i)3-s + (−1.90 + 0.618i)4-s + (−1.12 + 1.93i)5-s + (0.756 − 2.32i)6-s + (0.312 + 0.312i)7-s + (−1.28 − 2.52i)8-s + (1.76 + 2.42i)9-s + (−2.94 − 1.14i)10-s + (−4.08 − 2.96i)11-s + (3.42 + 0.541i)12-s + (−0.367 + 0.505i)14-s + (3.25 − 2.09i)15-s + (3.23 − 2.35i)16-s + (−3 + 2.99i)18-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.891 − 0.453i)3-s + (−0.951 + 0.309i)4-s + (−0.503 + 0.863i)5-s + (0.309 − 0.951i)6-s + (0.118 + 0.118i)7-s + (−0.453 − 0.891i)8-s + (0.587 + 0.809i)9-s + (−0.932 − 0.362i)10-s + (−1.23 − 0.894i)11-s + (0.987 + 0.156i)12-s + (−0.0982 + 0.135i)14-s + (0.840 − 0.541i)15-s + (0.809 − 0.587i)16-s + (−0.707 + 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.647 + 0.762i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.647 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360453 - 0.166793i\)
\(L(\frac12)\) \(\approx\) \(0.360453 - 0.166793i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.221 - 1.39i)T \)
3 \( 1 + (1.54 + 0.786i)T \)
5 \( 1 + (1.12 - 1.93i)T \)
good7 \( 1 + (-0.312 - 0.312i)T + 7iT^{2} \)
11 \( 1 + (4.08 + 2.96i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (-5.08 + 1.65i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.40 + 10.4i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (4.90 + 2.50i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (7.07 + 9.73i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (16.6 - 2.63i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (5.71 - 1.85i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.68 - 13.1i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (8.33 - 16.3i)T + (-57.0 - 78.4i)T^{2} \)
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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

−10.62459223913426025226773541420, −9.772979629712412538411290085121, −8.179135889794521042623427403813, −7.87817034251455184159784526931, −6.83416989246620621102459523871, −6.10220233934661726963506126528, −5.29666184529237495174469672653, −4.19844841410348504909158376516, −2.78657136508139348031345341117, −0.25862257868250705058159061965, 1.32258838176004630155984715683, 3.09598333981143800711123192882, 4.53825400194687121635534544264, 4.78212457935231248537617025479, 5.81128745063747023926422587952, 7.30437110854902658579351700625, 8.405325360254808446771018081583, 9.291580781071736789969273399794, 10.26122085890541836877070849195, 10.67767113546251873538441198940

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