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Properties

Label 2-600-120.53-c1-0-24
Degree $2$
Conductor $600$
Sign $0.989 - 0.146i$
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  + (0.178 + 1.40i)2-s + (−1.22 − 1.22i)3-s + (−1.93 + 0.5i)4-s + (1.5 − 1.93i)6-s + (−1.04 − 2.62i)8-s + 2.99i·9-s + (2.98 + 1.75i)12-s + (3.50 − 1.93i)16-s + (−3.16 − 3.16i)17-s + (−4.20 + 0.534i)18-s + 7.74·19-s + (6.32 − 6.32i)23-s + (−1.93 + 4.5i)24-s + (3.67 − 3.67i)27-s + 8·31-s + (3.34 + 4.56i)32-s + ⋯
L(s)  = 1  + (0.126 + 0.992i)2-s + (−0.707 − 0.707i)3-s + (−0.968 + 0.250i)4-s + (0.612 − 0.790i)6-s + (−0.370 − 0.929i)8-s + 0.999i·9-s + (0.861 + 0.507i)12-s + (0.875 − 0.484i)16-s + (−0.766 − 0.766i)17-s + (−0.992 + 0.126i)18-s + 1.77·19-s + (1.31 − 1.31i)23-s + (−0.395 + 0.918i)24-s + (0.707 − 0.707i)27-s + 1.43·31-s + (0.590 + 0.807i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.989 - 0.146i$
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: \((2,\ 600,\ (\ :1/2),\ 0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05496 + 0.0777841i\)
\(L(\frac12)\) \(\approx\) \(1.05496 + 0.0777841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.178 - 1.40i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (3.16 + 3.16i)T + 17iT^{2} \)
19 \( 1 - 7.74T + 19T^{2} \)
23 \( 1 + (-6.32 + 6.32i)T - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (6.32 + 6.32i)T + 47iT^{2} \)
53 \( 1 + (-9.79 - 9.79i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 15.4iT - 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 16iT - 79T^{2} \)
83 \( 1 + (-2.44 - 2.44i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−10.73009852482434333260872722446, −9.654574202919115984402661919473, −8.718611689843057222673221640609, −7.75604374092001356850588941290, −6.99291522231764822560024682752, −6.33978130901950911568901119356, −5.23844014303790244045494032078, −4.61534852005260949408034606767, −2.89788554579549637817478248834, −0.801214640041420657264453755360, 1.15311280498200147639066633316, 2.98470978950429566816199685167, 3.93987487354602016423689653805, 4.98953113490521539399984650163, 5.67061645657280291423551864772, 6.93247253439665083644703334384, 8.340184307364441731934559591853, 9.319710971456120737504550329332, 9.888711432707696446129807900324, 10.73230922500988653978253387651

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