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Properties

Label 2-600-120.59-c1-0-49
Degree $2$
Conductor $600$
Sign $0.625 + 0.780i$
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  + 1.41·2-s + (−1.57 − 0.724i)3-s + 2.00·4-s + (−2.22 − 1.02i)6-s + 2.82·8-s + (1.94 + 2.28i)9-s − 6.61i·11-s + (−3.14 − 1.44i)12-s + 4.00·16-s − 2.36·17-s + (2.75 + 3.22i)18-s + 8.34·19-s − 9.34i·22-s + (−4.44 − 2.04i)24-s + (−1.41 − 5.00i)27-s + ⋯
L(s)  = 1  + 1.00·2-s + (−0.908 − 0.418i)3-s + 1.00·4-s + (−0.908 − 0.418i)6-s + 1.00·8-s + (0.649 + 0.760i)9-s − 1.99i·11-s + (−0.908 − 0.418i)12-s + 1.00·16-s − 0.574·17-s + (0.649 + 0.760i)18-s + 1.91·19-s − 1.99i·22-s + (−0.908 − 0.418i)24-s + (−0.272 − 0.962i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.625 + 0.780i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94151 - 0.932316i\)
\(L(\frac12)\) \(\approx\) \(1.94151 - 0.932316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (1.57 + 0.724i)T \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 6.61iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 2.36T + 17T^{2} \)
19 \( 1 - 8.34T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 0.460iT - 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 - 12.7iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.98622227634956495363835605060, −10.04311608882989476607921437329, −8.597917335687251840120798474385, −7.54378062633750245293468182176, −6.71281979772240830373715386804, −5.72929523346046450112910755463, −5.31787255740072950070921517663, −3.95357851316239551897722462921, −2.82023621704821017627542107399, −1.10168728161086520050627443188, 1.70482468713934778770693159705, 3.30544441743702324893685626542, 4.59023768141134005300537684335, 4.95948004102051586294623536794, 6.12114083821065295590673616269, 6.99929171587923751199703895552, 7.67557971440937989648392295032, 9.541668086077248409636688239825, 9.966965179117804285442009654464, 11.06466196000536458592710308661

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