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Properties

Label 2-7200-5.4-c1-0-40
Degree $2$
Conductor $7200$
Sign $0.894 - 0.447i$
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  + 4·11-s + 2i·13-s − 2i·17-s + 8·19-s + 4i·23-s − 6·29-s − 2i·37-s + 6·41-s − 4i·43-s + 12i·47-s + 7·49-s + 6i·53-s − 12·59-s + 14·61-s − 12i·67-s + ⋯
L(s)  = 1  + 1.20·11-s + 0.554i·13-s − 0.485i·17-s + 1.83·19-s + 0.834i·23-s − 1.11·29-s − 0.328i·37-s + 0.937·41-s − 0.609i·43-s + 1.75i·47-s + 49-s + 0.824i·53-s − 1.56·59-s + 1.79·61-s − 1.46i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (6049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.322141582\)
\(L(\frac12)\) \(\approx\) \(2.322141582\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−7.66207430316753688748534729164, −7.44610161722978143922597859654, −6.61615527837999324447270766584, −5.84565615372494755067694723813, −5.23952919251081496661943120189, −4.29694640681980916209141146769, −3.66638445494003153698409979302, −2.86514723193796199652280684450, −1.73444234612731094140863624316, −0.932513889295250556769573727180, 0.72201391453829937284079479117, 1.61885372727862841894104865554, 2.71294551667721947694464722278, 3.58376127349263816501765651718, 4.14065374167586685781813738531, 5.16814582971103776287837552771, 5.72706747199245297884750902979, 6.51029837377020657996277402207, 7.19963041675976963399598464222, 7.79565758353505274496884020399

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