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Properties

Label 2-600-15.14-c2-0-10
Degree $2$
Conductor $600$
Sign $0.694 - 0.719i$
Analytic cond. $16.3488$
Root an. cond. $4.04336$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.82 + i)3-s + i·7-s + (7.00 − 5.65i)9-s + 8.48i·11-s − 15i·13-s − 19.7·17-s + 23·19-s + (−1 − 2.82i)21-s + 2.82·23-s + (−14.1 + 23.0i)27-s − 25.4i·29-s + 33·31-s + (−8.48 − 24i)33-s + 66i·37-s + (15 + 42.4i)39-s + ⋯
L(s)  = 1  + (−0.942 + 0.333i)3-s + 0.142i·7-s + (0.777 − 0.628i)9-s + 0.771i·11-s − 1.15i·13-s − 1.16·17-s + 1.21·19-s + (−0.0476 − 0.134i)21-s + 0.122·23-s + (−0.523 + 0.851i)27-s − 0.877i·29-s + 1.06·31-s + (−0.257 − 0.727i)33-s + 1.78i·37-s + (0.384 + 1.08i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.185345852\)
\(L(\frac12)\) \(\approx\) \(1.185345852\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.82 - i)T \)
5 \( 1 \)
good7 \( 1 - iT - 49T^{2} \)
11 \( 1 - 8.48iT - 121T^{2} \)
13 \( 1 + 15iT - 169T^{2} \)
17 \( 1 + 19.7T + 289T^{2} \)
19 \( 1 - 23T + 361T^{2} \)
23 \( 1 - 2.82T + 529T^{2} \)
29 \( 1 + 25.4iT - 841T^{2} \)
31 \( 1 - 33T + 961T^{2} \)
37 \( 1 - 66iT - 1.36e3T^{2} \)
41 \( 1 - 36.7iT - 1.68e3T^{2} \)
43 \( 1 - 7iT - 1.84e3T^{2} \)
47 \( 1 - 45.2T + 2.20e3T^{2} \)
53 \( 1 - 36.7T + 2.80e3T^{2} \)
59 \( 1 - 101. iT - 3.48e3T^{2} \)
61 \( 1 - 39T + 3.72e3T^{2} \)
67 \( 1 - 113iT - 4.48e3T^{2} \)
71 \( 1 - 25.4iT - 5.04e3T^{2} \)
73 \( 1 + 58iT - 5.32e3T^{2} \)
79 \( 1 + 70T + 6.24e3T^{2} \)
83 \( 1 - 152.T + 6.88e3T^{2} \)
89 \( 1 + 90.5iT - 7.92e3T^{2} \)
97 \( 1 + iT - 9.40e3T^{2} \)
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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

−10.40005503744847927946465747501, −9.998852435391264490099284032422, −8.975421607528937183184075246907, −7.80228342008721909223420777163, −6.88236647444888242835779850749, −5.94335014515877511759054732596, −5.03952795429279532778955661845, −4.19198524946147350491427206735, −2.74156102608161819553270175924, −0.962570980201565591014918383130, 0.67888393166939708375749859859, 2.13070535373146273632353802947, 3.81001793541494556200819869830, 4.88023712237873993451638235988, 5.82318548676376961588520460899, 6.75764504122393515130351182820, 7.40396228426564183059796231452, 8.665575629737583102906356634966, 9.479362193614405469591193654321, 10.64284902004319127269149960818

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