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Properties

Label 2-600-24.11-c1-0-24
Degree $2$
Conductor $600$
Sign $0.997 + 0.0748i$
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  + (−0.541 − 1.30i)2-s + (1.64 − 0.541i)3-s + (−1.41 + 1.41i)4-s + (−1.59 − 1.85i)6-s + 3.29i·7-s + (2.61 + 1.08i)8-s + (2.41 − 1.78i)9-s + 2.51i·11-s + (−1.56 + 3.09i)12-s + 4.65i·13-s + (4.29 − 1.78i)14-s − 4i·16-s + 3.69i·17-s + (−3.63 − 2.19i)18-s − 0.828·19-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (0.949 − 0.312i)3-s + (−0.707 + 0.707i)4-s + (−0.652 − 0.758i)6-s + 1.24i·7-s + (0.923 + 0.382i)8-s + (0.804 − 0.593i)9-s + 0.759i·11-s + (−0.450 + 0.892i)12-s + 1.29i·13-s + (1.14 − 0.475i)14-s − i·16-s + 0.896i·17-s + (−0.856 − 0.516i)18-s − 0.190·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.997 + 0.0748i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.997 + 0.0748i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52507 - 0.0571544i\)
\(L(\frac12)\) \(\approx\) \(1.52507 - 0.0571544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.541 + 1.30i)T \)
3 \( 1 + (-1.64 + 0.541i)T \)
5 \( 1 \)
good7 \( 1 - 3.29iT - 7T^{2} \)
11 \( 1 - 2.51iT - 11T^{2} \)
13 \( 1 - 4.65iT - 13T^{2} \)
17 \( 1 - 3.69iT - 17T^{2} \)
19 \( 1 + 0.828T + 19T^{2} \)
23 \( 1 + 2.61T + 23T^{2} \)
29 \( 1 - 6.08T + 29T^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 + 1.92iT - 37T^{2} \)
41 \( 1 + 8.59iT - 41T^{2} \)
43 \( 1 - 6.01T + 43T^{2} \)
47 \( 1 + 2.61T + 47T^{2} \)
53 \( 1 + 4.59T + 53T^{2} \)
59 \( 1 - 2.51iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 - 3.29T + 67T^{2} \)
71 \( 1 + 7.12T + 71T^{2} \)
73 \( 1 - 6.58T + 73T^{2} \)
79 \( 1 + 16.4iT - 79T^{2} \)
83 \( 1 - 9.37iT - 83T^{2} \)
89 \( 1 + 5.03iT - 89T^{2} \)
97 \( 1 - 2.72T + 97T^{2} \)
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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

−10.50636348174157885904174481170, −9.601541443591582951165741950895, −8.981278861396695093071728036003, −8.376291981765309227300961431583, −7.40434306945660048576071684065, −6.26455169353810552172600846858, −4.68205826024823204414672735318, −3.74350647082166049964313095924, −2.42320260286703554364015570929, −1.80382590912374269806675394530, 0.941185867691920606161999502970, 3.02727823447045523354146565948, 4.15892667876029274329909580970, 5.09747345076862720203935232064, 6.36806234638410291606278071449, 7.39756266381756792548476363367, 7.990338572354468834784413571928, 8.703258592419156321657502432522, 9.804113266166563254996105222806, 10.28498016070111642092620229889

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