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Properties

Label 2-2400-120.59-c1-0-0
Degree $2$
Conductor $2400$
Sign $-0.999 + 0.0319i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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  + (−1.57 − 0.724i)3-s + (1.94 + 2.28i)9-s + 6.61i·11-s + 2.36·17-s − 8.34·19-s + (−1.41 − 5.00i)27-s + (4.79 − 10.3i)33-s − 0.460i·41-s − 10i·43-s − 7·49-s + (−3.72 − 1.71i)51-s + (13.1 + 6.05i)57-s − 14.1i·59-s + 14.3i·67-s − 13.6i·73-s + ⋯
L(s)  = 1  + (−0.908 − 0.418i)3-s + (0.649 + 0.760i)9-s + 1.99i·11-s + 0.574·17-s − 1.91·19-s + (−0.272 − 0.962i)27-s + (0.833 − 1.81i)33-s − 0.0719i·41-s − 1.52i·43-s − 49-s + (−0.521 − 0.240i)51-s + (1.73 + 0.801i)57-s − 1.84i·59-s + 1.75i·67-s − 1.60i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-0.999 + 0.0319i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ -0.999 + 0.0319i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08304939242\)
\(L(\frac12)\) \(\approx\) \(0.08304939242\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
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

−9.532123035298096563669769895684, −8.453376923941937034871858496718, −7.64421385808327175117506039067, −6.93099237109056550554774744767, −6.41340660660268195265553598492, −5.39599357811044935962037602321, −4.66797087074622517353026846596, −3.97640931768406617251789032326, −2.33116980264953798994147476748, −1.62401183801837787451852164373, 0.03341534566370987926652358876, 1.24717418008420296692210151922, 2.86553092991111892812614172782, 3.78556967230874404445615896251, 4.58676897952497331191535240185, 5.60083732265888438148861432556, 6.11921465060296673122566232811, 6.73456779416261214030677473261, 7.951081418275001591187750434987, 8.583463561587991176512171200813

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