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Properties

Label 2-600-120.53-c1-0-38
Degree $2$
Conductor $600$
Sign $0.796 - 0.604i$
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  + (1.39 + 0.250i)2-s + (−1.40 + 1.01i)3-s + (1.87 + 0.696i)4-s + (−2.20 + 1.06i)6-s + (2.29 − 2.29i)7-s + (2.43 + 1.43i)8-s + (0.925 − 2.85i)9-s + 2.28·11-s + (−3.33 + 0.934i)12-s + (−1.05 + 1.05i)13-s + (3.76 − 2.61i)14-s + (3.03 + 2.61i)16-s + (3.04 + 3.04i)17-s + (2.00 − 3.74i)18-s − 3.36·19-s + ⋯
L(s)  = 1  + (0.984 + 0.176i)2-s + (−0.808 + 0.588i)3-s + (0.937 + 0.348i)4-s + (−0.900 + 0.435i)6-s + (0.865 − 0.865i)7-s + (0.861 + 0.508i)8-s + (0.308 − 0.951i)9-s + 0.688·11-s + (−0.962 + 0.269i)12-s + (−0.292 + 0.292i)13-s + (1.00 − 0.698i)14-s + (0.757 + 0.652i)16-s + (0.738 + 0.738i)17-s + (0.471 − 0.881i)18-s − 0.772·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.796 - 0.604i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28031 + 0.767094i\)
\(L(\frac12)\) \(\approx\) \(2.28031 + 0.767094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.250i)T \)
3 \( 1 + (1.40 - 1.01i)T \)
5 \( 1 \)
good7 \( 1 + (-2.29 + 2.29i)T - 7iT^{2} \)
11 \( 1 - 2.28T + 11T^{2} \)
13 \( 1 + (1.05 - 1.05i)T - 13iT^{2} \)
17 \( 1 + (-3.04 - 3.04i)T + 17iT^{2} \)
19 \( 1 + 3.36T + 19T^{2} \)
23 \( 1 + (-3.68 + 3.68i)T - 23iT^{2} \)
29 \( 1 - 2.71iT - 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 + (2.31 + 2.31i)T + 37iT^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + (-1.16 + 1.16i)T - 43iT^{2} \)
47 \( 1 + (1.83 + 1.83i)T + 47iT^{2} \)
53 \( 1 + (-5.82 - 5.82i)T + 53iT^{2} \)
59 \( 1 + 7.41iT - 59T^{2} \)
61 \( 1 + 8.97iT - 61T^{2} \)
67 \( 1 + (8.66 + 8.66i)T + 67iT^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + (1.83 + 1.83i)T + 73iT^{2} \)
79 \( 1 + 8.28iT - 79T^{2} \)
83 \( 1 + (5.27 + 5.27i)T + 83iT^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (-2.79 + 2.79i)T - 97iT^{2} \)
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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

−10.84656243788467192938985391807, −10.37784303131898496694326609089, −9.049524725132086777060633856538, −7.83008564491841123702067362898, −6.88911968985600872792175932196, −6.11910287674827187430620193286, −5.00461270621509933999423523218, −4.35849185847485191657747462082, −3.48587094585095014731697363672, −1.52868058249336616803564317838, 1.42956657630246152250154703489, 2.56560723038192681091682203296, 4.13653735126115969391079007494, 5.34266956015243815911833573272, 5.60991473068035789106510923192, 6.87670074437146794385502398693, 7.53412778337532644745893017323, 8.731015915736194641586107576307, 10.02815953421876637420835401337, 11.00826624039516103111478540149

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