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Properties

Label 2-2400-20.3-c1-0-10
Degree $2$
Conductor $2400$
Sign $-0.130 - 0.991i$
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  + (−0.707 + 0.707i)3-s + (2.02 + 2.02i)7-s − 1.00i·9-s + 0.964i·11-s + (2.63 + 2.63i)13-s + (−2.14 + 2.14i)17-s + 4.76·19-s − 2.86·21-s + (0.282 − 0.282i)23-s + (0.707 + 0.707i)27-s − 7.32i·29-s + 8.42i·31-s + (−0.682 − 0.682i)33-s + (1.36 − 1.36i)37-s − 3.73·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.765 + 0.765i)7-s − 0.333i·9-s + 0.290i·11-s + (0.731 + 0.731i)13-s + (−0.520 + 0.520i)17-s + 1.09·19-s − 0.624·21-s + (0.0589 − 0.0589i)23-s + (0.136 + 0.136i)27-s − 1.36i·29-s + 1.51i·31-s + (−0.118 − 0.118i)33-s + (0.224 − 0.224i)37-s − 0.597·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ -0.130 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.627724545\)
\(L(\frac12)\) \(\approx\) \(1.627724545\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-2.02 - 2.02i)T + 7iT^{2} \)
11 \( 1 - 0.964iT - 11T^{2} \)
13 \( 1 + (-2.63 - 2.63i)T + 13iT^{2} \)
17 \( 1 + (2.14 - 2.14i)T - 17iT^{2} \)
19 \( 1 - 4.76T + 19T^{2} \)
23 \( 1 + (-0.282 + 0.282i)T - 23iT^{2} \)
29 \( 1 + 7.32iT - 29T^{2} \)
31 \( 1 - 8.42iT - 31T^{2} \)
37 \( 1 + (-1.36 + 1.36i)T - 37iT^{2} \)
41 \( 1 + 8.05T + 41T^{2} \)
43 \( 1 + (-3.30 + 3.30i)T - 43iT^{2} \)
47 \( 1 + (-6.87 - 6.87i)T + 47iT^{2} \)
53 \( 1 + (-2.33 - 2.33i)T + 53iT^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + (-1.03 - 1.03i)T + 67iT^{2} \)
71 \( 1 + 4.84iT - 71T^{2} \)
73 \( 1 + (-9.46 - 9.46i)T + 73iT^{2} \)
79 \( 1 - 1.53T + 79T^{2} \)
83 \( 1 + (-8.94 + 8.94i)T - 83iT^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 + (12.9 - 12.9i)T - 97iT^{2} \)
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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

−9.109587388002207415374009567479, −8.550952667959818104051293788125, −7.70425837922569864256303251298, −6.73759246670143191613353210760, −5.97692606764130842918892222319, −5.21161474048947243602878521462, −4.47940038245523229359706844162, −3.60392714401025432454597340674, −2.36400833668687167739956217479, −1.31262761546102251774001525416, 0.64744500825669445950105949114, 1.58569273204174491172202506610, 2.93897484548196531075279406672, 3.92013008955854510557077856690, 4.91907921185091403721945142420, 5.56466683135946266536578811109, 6.47136532343833537503725324418, 7.33013673728692828173426650045, 7.83308326497093113914055202504, 8.629390689930910837817631360448

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