Properties

Label 2-1200-5.4-c3-0-34
Degree $2$
Conductor $1200$
Sign $0.447 + 0.894i$
Analytic cond. $70.8022$
Root an. cond. $8.41440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 23i·7-s − 9·9-s + 30·11-s − 29i·13-s + 78i·17-s + 149·19-s − 69·21-s + 150i·23-s + 27i·27-s + 234·29-s + 217·31-s − 90i·33-s + 146i·37-s − 87·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.24i·7-s − 0.333·9-s + 0.822·11-s − 0.618i·13-s + 1.11i·17-s + 1.79·19-s − 0.717·21-s + 1.35i·23-s + 0.192i·27-s + 1.49·29-s + 1.25·31-s − 0.474i·33-s + 0.648i·37-s − 0.357·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(70.8022\)
Root analytic conductor: \(8.41440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.454059046\)
\(L(\frac12)\) \(\approx\) \(2.454059046\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
5 \( 1 \)
good7 \( 1 + 23iT - 343T^{2} \)
11 \( 1 - 30T + 1.33e3T^{2} \)
13 \( 1 + 29iT - 2.19e3T^{2} \)
17 \( 1 - 78iT - 4.91e3T^{2} \)
19 \( 1 - 149T + 6.85e3T^{2} \)
23 \( 1 - 150iT - 1.21e4T^{2} \)
29 \( 1 - 234T + 2.43e4T^{2} \)
31 \( 1 - 217T + 2.97e4T^{2} \)
37 \( 1 - 146iT - 5.06e4T^{2} \)
41 \( 1 + 156T + 6.89e4T^{2} \)
43 \( 1 + 433iT - 7.95e4T^{2} \)
47 \( 1 + 30iT - 1.03e5T^{2} \)
53 \( 1 - 552iT - 1.48e5T^{2} \)
59 \( 1 + 270T + 2.05e5T^{2} \)
61 \( 1 - 275T + 2.26e5T^{2} \)
67 \( 1 + 803iT - 3.00e5T^{2} \)
71 \( 1 + 660T + 3.57e5T^{2} \)
73 \( 1 - 646iT - 3.89e5T^{2} \)
79 \( 1 - 992T + 4.93e5T^{2} \)
83 \( 1 + 846iT - 5.71e5T^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + 319iT - 9.12e5T^{2} \)
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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

−9.249577113901264539004440589836, −8.196902246826663031048149774595, −7.57317961276147072413263548143, −6.83410563506039453038955461420, −6.02049436124790095979914530722, −4.96989512164498237753778498928, −3.84498265680341999375986738709, −3.09055219644516106093369298991, −1.45386534155897433924503298068, −0.817735468916531309451112835020, 0.926552399898702445190354227642, 2.44861461137075857299241436782, 3.20553261783089610249821696390, 4.51988345346468985305412357483, 5.12781576271603856965176310994, 6.17008672141636631960439611969, 6.87522044329591554760566656315, 8.100817887404686094556285228167, 8.870337955392593314740584989692, 9.470790463770684221813329414928

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