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Properties

Label 2-7200-24.11-c1-0-26
Degree $2$
Conductor $7200$
Sign $-0.577 - 0.816i$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
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  + 1.16i·7-s + 5.88i·11-s + 7.16i·13-s + 6.32·19-s + 4.47·23-s + 4.83i·37-s + 7.53i·41-s + 2.82·47-s + 5.64·49-s − 5.65·53-s − 14.3i·59-s − 6.84·77-s + 0.955i·89-s − 8.32·91-s + 10.8i·103-s + ⋯
L(s)  = 1  + 0.439i·7-s + 1.77i·11-s + 1.98i·13-s + 1.45·19-s + 0.932·23-s + 0.795i·37-s + 1.17i·41-s + 0.412·47-s + 0.807·49-s − 0.777·53-s − 1.87i·59-s − 0.779·77-s + 0.101i·89-s − 0.872·91-s + 1.06i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (4751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.958980551\)
\(L(\frac12)\) \(\approx\) \(1.958980551\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.16iT - 7T^{2} \)
11 \( 1 - 5.88iT - 11T^{2} \)
13 \( 1 - 7.16iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 4.83iT - 37T^{2} \)
41 \( 1 - 7.53iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 0.955iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.106624703531957784918761376416, −7.29158269603293032932414731965, −6.89158210307475864297074272822, −6.22787212069520035543297185697, −5.09613637117293287683166791597, −4.73772496652406508831855407974, −3.95405589006433214556956910091, −2.92806121343587577924546696891, −2.03650039024361921020804367614, −1.34959746873776101250134503787, 0.55320762096330501495503145497, 1.07649899499282530768635670748, 2.71361123677949582996071874687, 3.23036706415845306094317535913, 3.85188378694569308015151013493, 5.07368824960616495527535222945, 5.61789640643300752433177058435, 6.04483196102791743356256814145, 7.25307856291347596164714581171, 7.56270927954416519651440883513

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