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Properties

Label 2-600-40.29-c1-0-20
Degree $2$
Conductor $600$
Sign $-0.0534 + 0.998i$
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  + (−0.192 − 1.40i)2-s + 3-s + (−1.92 + 0.540i)4-s + (−0.192 − 1.40i)6-s + 0.0802i·7-s + (1.12 + 2.59i)8-s + 9-s − 2.41i·11-s + (−1.92 + 0.540i)12-s + 5.26·13-s + (0.112 − 0.0154i)14-s + (3.41 − 2.08i)16-s − 0.255i·17-s + (−0.192 − 1.40i)18-s − 6.95i·19-s + ⋯
L(s)  = 1  + (−0.136 − 0.990i)2-s + 0.577·3-s + (−0.962 + 0.270i)4-s + (−0.0786 − 0.571i)6-s + 0.0303i·7-s + (0.398 + 0.917i)8-s + 0.333·9-s − 0.728i·11-s + (−0.555 + 0.155i)12-s + 1.46·13-s + (0.0300 − 0.00413i)14-s + (0.854 − 0.520i)16-s − 0.0620i·17-s + (−0.0454 − 0.330i)18-s − 1.59i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0534 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0534 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.0534 + 0.998i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.0534 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08530 - 1.14491i\)
\(L(\frac12)\) \(\approx\) \(1.08530 - 1.14491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.192 + 1.40i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 - 0.0802iT - 7T^{2} \)
11 \( 1 + 2.41iT - 11T^{2} \)
13 \( 1 - 5.26T + 13T^{2} \)
17 \( 1 + 0.255iT - 17T^{2} \)
19 \( 1 + 6.95iT - 19T^{2} \)
23 \( 1 - 1.64iT - 23T^{2} \)
29 \( 1 + 4.51iT - 29T^{2} \)
31 \( 1 - 8.29T + 31T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 + 8.11T + 41T^{2} \)
43 \( 1 + 4.08T + 43T^{2} \)
47 \( 1 - 5.70iT - 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 12.6iT - 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 - 7.27T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + 5.50T + 79T^{2} \)
83 \( 1 + 9.20T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 8.50iT - 97T^{2} \)
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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

−10.48292051307635595490312000778, −9.611230847598259294857623591265, −8.604968724311367758345699398429, −8.378246667555335860730713855414, −6.97567398769202417744445317644, −5.71491856061221867009397865797, −4.47030542909373964019411101685, −3.48828450908188872980928251306, −2.55970773399784185044009798416, −1.03607698790337844962063647822, 1.51253127233738562753896060535, 3.46458769370075850693271056108, 4.35262593059949547487455998694, 5.56847571645537210547789287256, 6.50661125138962935987282233594, 7.35373577431125166347434900557, 8.420820675020302206770699887227, 8.701346808831228791517399726819, 10.01088735938521367256414883593, 10.39055207846927220143904272302

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