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Properties

Label 2-7200-40.29-c1-0-40
Degree $2$
Conductor $7200$
Sign $0.994 + 0.102i$
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  − 2i·7-s + 2i·11-s − 5.29i·19-s + 5.29i·23-s + 6i·29-s − 4·31-s − 10.5·37-s + 10.5·41-s + 10.5·43-s − 5.29i·47-s + 3·49-s + 2·53-s − 10i·59-s + 10.5i·61-s + 10.5·71-s + ⋯
L(s)  = 1  − 0.755i·7-s + 0.603i·11-s − 1.21i·19-s + 1.10i·23-s + 1.11i·29-s − 0.718·31-s − 1.73·37-s + 1.65·41-s + 1.61·43-s − 0.771i·47-s + 0.428·49-s + 0.274·53-s − 1.30i·59-s + 1.35i·61-s + 1.25·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.994 + 0.102i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 0.994 + 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.853436293\)
\(L(\frac12)\) \(\approx\) \(1.853436293\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 - 5.29iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 5.29iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−7.70911467152421228517695453941, −7.13552170148386202053465951619, −6.83767143038572209381550463093, −5.66346438073431523850221835093, −5.14183725723257532345681819953, −4.24979806806465041414669966312, −3.66236926964035157377966023732, −2.70036645238216543222959981614, −1.74346427855236230104238792028, −0.70279904433599758660255375424, 0.67536781244110429537170552294, 1.94379237936526362997946633075, 2.64952880922600794610705034496, 3.59693908041954476370488146861, 4.29205658391621457955281615903, 5.23468506567377163099470193623, 5.95955021508237611101014945839, 6.27264724927916123932463794072, 7.38821843763330444304316535280, 7.912610982343659384993579381312

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