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Properties

Label 2-7200-12.11-c1-0-50
Degree $2$
Conductor $7200$
Sign $0.985 + 0.169i$
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  + 4i·7-s + 5.65·11-s − 4·13-s − 4.24i·17-s + 5.65·23-s − 1.41i·29-s − 4i·31-s + 6·37-s − 9.89i·41-s − 8i·43-s + 5.65·47-s − 9·49-s − 4.24i·53-s − 11.3·59-s − 2·61-s + ⋯
L(s)  = 1  + 1.51i·7-s + 1.70·11-s − 1.10·13-s − 1.02i·17-s + 1.17·23-s − 0.262i·29-s − 0.718i·31-s + 0.986·37-s − 1.54i·41-s − 1.21i·43-s + 0.825·47-s − 1.28·49-s − 0.582i·53-s − 1.47·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.134829727\)
\(L(\frac12)\) \(\approx\) \(2.134829727\)
\(L(\frac{3}{2})\) 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 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + 4.24iT - 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−7.86451892107228650430172149398, −7.13892181054178486410959935838, −6.53690300391490128096231427208, −5.76503358037135691773311641731, −5.13706109999569795012872379302, −4.41993708966906554351307549509, −3.45888550948050011686097142701, −2.57720302042661161888157117833, −1.96332553051199315458998853228, −0.64080193492097473913536882666, 0.944864860667145704091545992116, 1.52376312349630613158836103125, 2.89005201282552288741082975291, 3.67643302055778714449512818796, 4.40149108470166146262312500398, 4.81520140010966512422824404706, 6.10271850288060723933067686280, 6.57972749124769553097936350347, 7.27383040058537241223590082964, 7.73326349156638845955956084043

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