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Properties

Label 2-600-120.53-c1-0-27
Degree $2$
Conductor $600$
Sign $-0.402 - 0.915i$
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  + (0.722 + 1.21i)2-s + (1.48 + 0.889i)3-s + (−0.956 + 1.75i)4-s + (−0.00827 + 2.44i)6-s + (2.09 − 2.09i)7-s + (−2.82 + 0.106i)8-s + (1.41 + 2.64i)9-s + 2.65·11-s + (−2.98 + 1.75i)12-s + (−4.21 + 4.21i)13-s + (4.05 + 1.03i)14-s + (−2.17 − 3.35i)16-s + (3.84 + 3.84i)17-s + (−2.19 + 3.63i)18-s − 3.15·19-s + ⋯
L(s)  = 1  + (0.510 + 0.859i)2-s + (0.857 + 0.513i)3-s + (−0.478 + 0.878i)4-s + (−0.00337 + 0.999i)6-s + (0.790 − 0.790i)7-s + (−0.999 + 0.0377i)8-s + (0.472 + 0.881i)9-s + 0.800·11-s + (−0.861 + 0.507i)12-s + (−1.16 + 1.16i)13-s + (1.08 + 0.275i)14-s + (−0.542 − 0.839i)16-s + (0.933 + 0.933i)17-s + (−0.516 + 0.856i)18-s − 0.724·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.402 - 0.915i$
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: \((2,\ 600,\ (\ :1/2),\ -0.402 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37350 + 2.10549i\)
\(L(\frac12)\) \(\approx\) \(1.37350 + 2.10549i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.722 - 1.21i)T \)
3 \( 1 + (-1.48 - 0.889i)T \)
5 \( 1 \)
good7 \( 1 + (-2.09 + 2.09i)T - 7iT^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
13 \( 1 + (4.21 - 4.21i)T - 13iT^{2} \)
17 \( 1 + (-3.84 - 3.84i)T + 17iT^{2} \)
19 \( 1 + 3.15T + 19T^{2} \)
23 \( 1 + (-1.60 + 1.60i)T - 23iT^{2} \)
29 \( 1 + 4.35iT - 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + (4.94 + 4.94i)T + 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (0.219 - 0.219i)T - 43iT^{2} \)
47 \( 1 + (1.83 + 1.83i)T + 47iT^{2} \)
53 \( 1 + (-4.64 - 4.64i)T + 53iT^{2} \)
59 \( 1 + 9.93iT - 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + (-8.80 - 8.80i)T + 67iT^{2} \)
71 \( 1 - 2.89iT - 71T^{2} \)
73 \( 1 + (2.29 + 2.29i)T + 73iT^{2} \)
79 \( 1 + 9.02iT - 79T^{2} \)
83 \( 1 + (-9.78 - 9.78i)T + 83iT^{2} \)
89 \( 1 + 2.83T + 89T^{2} \)
97 \( 1 + (9.26 - 9.26i)T - 97iT^{2} \)
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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

−10.85339832048937570680140388374, −9.845017959128416742051289225129, −9.003342184456735039293309281141, −8.193850783692815613302761201439, −7.41689347911728364511447324986, −6.63973299868666004993828422767, −5.20692611258658732842129839743, −4.28113249725699454488018098171, −3.76040749083558526084795364563, −2.10209857599959270804201153871, 1.26817375514035195943794751997, 2.50864994288882277320252463593, 3.29738123124466854865871652168, 4.69858528289618867850927075608, 5.53273934488272997088181826762, 6.80096206319519486320747485966, 7.921454346188344733720487694061, 8.754107279090394454886317256332, 9.570410888808352646453089149324, 10.32176384623315644253221423189

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