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Properties

Label 2-600-120.53-c1-0-22
Degree $2$
Conductor $600$
Sign $0.00866 + 0.999i$
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  + (−1.30 − 0.552i)2-s + (−1.65 + 0.504i)3-s + (1.38 + 1.43i)4-s + (2.43 + 0.257i)6-s + (−2.83 + 2.83i)7-s + (−1.01 − 2.64i)8-s + (2.48 − 1.67i)9-s − 4.74·11-s + (−3.02 − 1.68i)12-s + (0.867 − 0.867i)13-s + (5.26 − 2.12i)14-s + (−0.136 + 3.99i)16-s + (1.73 + 1.73i)17-s + (−4.16 + 0.803i)18-s − 3.35·19-s + ⋯
L(s)  = 1  + (−0.920 − 0.390i)2-s + (−0.956 + 0.291i)3-s + (0.694 + 0.719i)4-s + (0.994 + 0.105i)6-s + (−1.07 + 1.07i)7-s + (−0.358 − 0.933i)8-s + (0.829 − 0.557i)9-s − 1.43·11-s + (−0.874 − 0.485i)12-s + (0.240 − 0.240i)13-s + (1.40 − 0.568i)14-s + (−0.0341 + 0.999i)16-s + (0.419 + 0.419i)17-s + (−0.981 + 0.189i)18-s − 0.769·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00866 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00866 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.00866 + 0.999i$
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: \((2,\ 600,\ (\ :1/2),\ 0.00866 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212310 - 0.210478i\)
\(L(\frac12)\) \(\approx\) \(0.212310 - 0.210478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.552i)T \)
3 \( 1 + (1.65 - 0.504i)T \)
5 \( 1 \)
good7 \( 1 + (2.83 - 2.83i)T - 7iT^{2} \)
11 \( 1 + 4.74T + 11T^{2} \)
13 \( 1 + (-0.867 + 0.867i)T - 13iT^{2} \)
17 \( 1 + (-1.73 - 1.73i)T + 17iT^{2} \)
19 \( 1 + 3.35T + 19T^{2} \)
23 \( 1 + (-4.82 + 4.82i)T - 23iT^{2} \)
29 \( 1 + 0.936iT - 29T^{2} \)
31 \( 1 - 5.49T + 31T^{2} \)
37 \( 1 + (0.749 + 0.749i)T + 37iT^{2} \)
41 \( 1 - 2.24iT - 41T^{2} \)
43 \( 1 + (-8.75 + 8.75i)T - 43iT^{2} \)
47 \( 1 + (6.71 + 6.71i)T + 47iT^{2} \)
53 \( 1 + (8.46 + 8.46i)T + 53iT^{2} \)
59 \( 1 + 6.53iT - 59T^{2} \)
61 \( 1 + 5.10iT - 61T^{2} \)
67 \( 1 + (-4.85 - 4.85i)T + 67iT^{2} \)
71 \( 1 + 8.34iT - 71T^{2} \)
73 \( 1 + (1.57 + 1.57i)T + 73iT^{2} \)
79 \( 1 + 7.75iT - 79T^{2} \)
83 \( 1 + (-9.15 - 9.15i)T + 83iT^{2} \)
89 \( 1 + 4.97T + 89T^{2} \)
97 \( 1 + (-1.42 + 1.42i)T - 97iT^{2} \)
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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

−10.41784971457055874768063649667, −9.790262334343009178497796173087, −8.850940503199146047859119773038, −7.998916081111693023362849230130, −6.72858797856428330887855276500, −6.08722516088602410891415848393, −5.02490751345545767310826269190, −3.44508098640655981220073397510, −2.36855404759552667757052655691, −0.30142082199738274936717819484, 1.05508455614442537144995014398, 2.87644608733632736770254580101, 4.62597715540803744812071622939, 5.72474872720645253092023132275, 6.54257477069795854376229497939, 7.32930690501148078154064079207, 7.938233923104073687440077927638, 9.355805979114473501842064360425, 10.10005704082001030541240244363, 10.72253552029654999125805896177

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