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Properties

Label 2-7200-8.5-c1-0-44
Degree $2$
Conductor $7200$
Sign $0.965 - 0.258i$
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  + 0.732·7-s + 2i·11-s − 3.46i·13-s + 3.46·17-s + 0.535i·19-s + 6.19·23-s + 6.92i·29-s + 5.46·31-s − 2i·37-s − 1.46·41-s + 5.26i·43-s − 3.26·47-s − 6.46·49-s − 11.4i·53-s + 7.46i·59-s + ⋯
L(s)  = 1  + 0.276·7-s + 0.603i·11-s − 0.960i·13-s + 0.840·17-s + 0.122i·19-s + 1.29·23-s + 1.28i·29-s + 0.981·31-s − 0.328i·37-s − 0.228·41-s + 0.803i·43-s − 0.476·47-s − 0.923·49-s − 1.57i·53-s + 0.971i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.965 - 0.258i$
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: \((2,\ 7200,\ (\ :1/2),\ 0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.200943211\)
\(L(\frac12)\) \(\approx\) \(2.200943211\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 0.535iT - 19T^{2} \)
23 \( 1 - 6.19T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 - 5.26iT - 43T^{2} \)
47 \( 1 + 3.26T + 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 - 7.46iT - 59T^{2} \)
61 \( 1 - 8.92iT - 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + 7.46T + 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 - 1.26iT - 83T^{2} \)
89 \( 1 + 8.92T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−7.950191866537723788141299933678, −7.28459336672672253973631857986, −6.65737473069404178889023875957, −5.75934663024297788387983194931, −5.08407876486799013933822311377, −4.55985690545728024525193751384, −3.39685060049567578820883270883, −2.92421014296126352813731410517, −1.74409181101807960840247744029, −0.844582828836337705682473806206, 0.72127993693250806088246944841, 1.70296722530321838429599859013, 2.73830142421368923380013122667, 3.47763395559296021226023896024, 4.39513782897119132437353519885, 5.00022631833351081218365192928, 5.84829873108416047859093253909, 6.49807271510975320049162760828, 7.18328110436812171166681736976, 7.961456150166893298237478978601

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