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Properties

Label 2-600-120.59-c1-0-59
Degree $2$
Conductor $600$
Sign $0.363 + 0.931i$
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  + 1.41·2-s + (0.158 − 1.72i)3-s + 2.00·4-s + (0.224 − 2.43i)6-s + 2.82·8-s + (−2.94 − 0.548i)9-s − 3.78i·11-s + (0.317 − 3.44i)12-s + 4.00·16-s + 8.02·17-s + (−4.17 − 0.775i)18-s − 6.34·19-s − 5.34i·22-s + (0.449 − 4.87i)24-s + (−1.41 + 4.99i)27-s + ⋯
L(s)  = 1  + 1.00·2-s + (0.0917 − 0.995i)3-s + 1.00·4-s + (0.0917 − 0.995i)6-s + 1.00·8-s + (−0.983 − 0.182i)9-s − 1.14i·11-s + (0.0917 − 0.995i)12-s + 1.00·16-s + 1.94·17-s + (−0.983 − 0.182i)18-s − 1.45·19-s − 1.14i·22-s + (0.0917 − 0.995i)24-s + (−0.272 + 0.962i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.363 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31496 - 1.58210i\)
\(L(\frac12)\) \(\approx\) \(2.31496 - 1.58210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (-0.158 + 1.72i)T \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 3.78iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 8.02T + 17T^{2} \)
19 \( 1 + 6.34T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 0.348iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 - 18.4iT - 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.92761361546323462479552180219, −9.743821548689322049846889530495, −8.244204211852060524778353350717, −7.87489883337886331600358625355, −6.58896977129318566933731144079, −6.05889978861569409091956936118, −5.09895747120993626699378297453, −3.61363964853908879114122741301, −2.74143437256550461782343291545, −1.29289675084234784794956700419, 2.12110011286142369096828815213, 3.40057104652322410010391382392, 4.25347043588027366066623516250, 5.15080911034242369683319675936, 5.96772288817883934493606935454, 7.15896328861654407693891262568, 8.087644300642154540922779430934, 9.268309265228042016128383461710, 10.35654556972212758495175541313, 10.57538453776370292412514980421

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