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Properties

Label 2-7200-8.5-c1-0-49
Degree $2$
Conductor $7200$
Sign $0.973 - 0.227i$
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  + 4.05·7-s + 0.985i·11-s − 4.94i·13-s − 4.52·17-s + 2.60i·19-s + 3.53·23-s + 7.59i·29-s + 3.28·31-s − 0.945i·37-s − 0.568·41-s + 8.45i·43-s + 2.60·47-s + 9.45·49-s + 0.229i·53-s + 9.10i·59-s + ⋯
L(s)  = 1  + 1.53·7-s + 0.297i·11-s − 1.37i·13-s − 1.09·17-s + 0.597i·19-s + 0.737·23-s + 1.41i·29-s + 0.589·31-s − 0.155i·37-s − 0.0887·41-s + 1.29i·43-s + 0.379·47-s + 1.35·49-s + 0.0315i·53-s + 1.18i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.973 - 0.227i$
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: \((2,\ 7200,\ (\ :1/2),\ 0.973 - 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.464374146\)
\(L(\frac12)\) \(\approx\) \(2.464374146\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.05T + 7T^{2} \)
11 \( 1 - 0.985iT - 11T^{2} \)
13 \( 1 + 4.94iT - 13T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 - 2.60iT - 19T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 - 7.59iT - 29T^{2} \)
31 \( 1 - 3.28T + 31T^{2} \)
37 \( 1 + 0.945iT - 37T^{2} \)
41 \( 1 + 0.568T + 41T^{2} \)
43 \( 1 - 8.45iT - 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 - 0.229iT - 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 - 8.45iT - 67T^{2} \)
71 \( 1 - 1.43T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 3.28T + 79T^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 3.23T + 97T^{2} \)
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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

−7.939138005660799139331715620755, −7.42621874042213894450422715094, −6.59713062161234715336160734784, −5.73154594122667630887345161326, −4.97748835140344879704384994292, −4.63048073355933985123804068476, −3.58959858653979556827824753739, −2.68139056820766230298417150644, −1.78030895435483270164284021459, −0.899834368225723499885599438790, 0.74736496101123241586873840856, 1.91649339030580286282588749338, 2.37782746456367092972813618468, 3.68815502697281139125041732455, 4.56277495216015349171554571725, 4.78950544978532369024934267367, 5.77772864964858976445057921575, 6.62953699805548477736160472314, 7.14883211508571106121968090260, 7.990566816396220038773035820439

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