Properties

Label 2-7200-8.5-c1-0-50
Degree $2$
Conductor $7200$
Sign $0.826 - 0.563i$
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  + 4.72·7-s − 3.93i·11-s + 3.46i·13-s + 3.51·17-s + 5.44i·19-s + 7.11·23-s + 3.66i·29-s − 5.23·31-s + 0.414i·37-s − 3.00·41-s + 5.34i·43-s + 0.925·47-s + 15.3·49-s − 0.233i·53-s + 14.3i·59-s + ⋯
L(s)  = 1  + 1.78·7-s − 1.18i·11-s + 0.961i·13-s + 0.852·17-s + 1.24i·19-s + 1.48·23-s + 0.681i·29-s − 0.940·31-s + 0.0681i·37-s − 0.469·41-s + 0.815i·43-s + 0.135·47-s + 2.18·49-s − 0.0320i·53-s + 1.87i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.826 - 0.563i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (3601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 0.826 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.763514326\)
\(L(\frac12)\) \(\approx\) \(2.763514326\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.72T + 7T^{2} \)
11 \( 1 + 3.93iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 3.51T + 17T^{2} \)
19 \( 1 - 5.44iT - 19T^{2} \)
23 \( 1 - 7.11T + 23T^{2} \)
29 \( 1 - 3.66iT - 29T^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 - 0.414iT - 37T^{2} \)
41 \( 1 + 3.00T + 41T^{2} \)
43 \( 1 - 5.34iT - 43T^{2} \)
47 \( 1 - 0.925T + 47T^{2} \)
53 \( 1 + 0.233iT - 53T^{2} \)
59 \( 1 - 14.3iT - 59T^{2} \)
61 \( 1 + 0.118iT - 61T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 - 0.563T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + 8.88T + 89T^{2} \)
97 \( 1 + 7.27T + 97T^{2} \)
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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

−7.944399572236503767885195698662, −7.47431101510184657298637849291, −6.64602603961541153511644006125, −5.63529763448764693926389633774, −5.31099418527664057115999576566, −4.43474579165290600558039672627, −3.70899439638878887139888318845, −2.80787782661585174411369540610, −1.62416351400831302676995478407, −1.15096389018395017848759825043, 0.74739031576930957479118405651, 1.72248972396766007845436634087, 2.48005389261664018262695053504, 3.50044678053235752031985391657, 4.48249320097736013325881573777, 5.16932220359920368849545923167, 5.31897874348328413756524346150, 6.62819890862945486996859493266, 7.34017046387391200473015682063, 7.79115768999110016923829862956

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