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Properties

Label 2-600-40.29-c1-0-21
Degree $2$
Conductor $600$
Sign $0.766 + 0.641i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.766 + 0.641i$
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: \((2,\ 600,\ (\ :1/2),\ 0.766 + 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634681 - 0.230579i\)
\(L(\frac12)\) \(\approx\) \(0.634681 - 0.230579i\)
\(L(\frac{3}{2})\) 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

−10.41528573415327318381951766247, −9.602636136027772536539928122874, −8.705796652773342602233232319232, −7.72233659688665529442913425804, −6.89121901097284999559661634175, −6.11008374001671873279655624261, −5.05671091149576887048236805366, −4.41175275965834762692367783102, −2.85090310150836847616522388435, −0.39865249560478206022760976376, 1.54654888827323749865831004164, 2.84501346881526334164471579589, 4.22677373175138084900531837283, 4.99591608274929784726825121237, 5.96543818906426696969132675634, 7.25088598625747830100063122968, 8.202550574922483207906573673715, 9.476407909465143944811424501526, 10.01661547115051536292808071021, 10.69303216697405152618555414971

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