Properties

Label 2-7728-1.1-c1-0-0
Degree $2$
Conductor $7728$
Sign $1$
Analytic cond. $61.7083$
Root an. cond. $7.85546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.26·5-s − 7-s + 9-s + 1.38·11-s − 3.69·13-s + 3.26·15-s − 4.95·17-s − 4.43·19-s + 21-s + 23-s + 5.64·25-s − 27-s − 6.66·29-s − 6.95·31-s − 1.38·33-s + 3.26·35-s + 6.66·37-s + 3.69·39-s + 6.66·41-s − 1.35·43-s − 3.26·45-s − 11.6·47-s + 49-s + 4.95·51-s − 9.06·53-s − 4.50·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.45·5-s − 0.377·7-s + 0.333·9-s + 0.416·11-s − 1.02·13-s + 0.842·15-s − 1.20·17-s − 1.01·19-s + 0.218·21-s + 0.208·23-s + 1.12·25-s − 0.192·27-s − 1.23·29-s − 1.24·31-s − 0.240·33-s + 0.551·35-s + 1.09·37-s + 0.591·39-s + 1.04·41-s − 0.206·43-s − 0.486·45-s − 1.69·47-s + 0.142·49-s + 0.693·51-s − 1.24·53-s − 0.607·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7728\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(61.7083\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1286719479\)
\(L(\frac12)\) \(\approx\) \(0.1286719479\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 + 3.26T + 5T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + 3.69T + 13T^{2} \)
17 \( 1 + 4.95T + 17T^{2} \)
19 \( 1 + 4.43T + 19T^{2} \)
29 \( 1 + 6.66T + 29T^{2} \)
31 \( 1 + 6.95T + 31T^{2} \)
37 \( 1 - 6.66T + 37T^{2} \)
41 \( 1 - 6.66T + 41T^{2} \)
43 \( 1 + 1.35T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 9.06T + 53T^{2} \)
59 \( 1 - 4.78T + 59T^{2} \)
61 \( 1 + 7.92T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 5.90T + 73T^{2} \)
79 \( 1 + 1.38T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + 2.11T + 89T^{2} \)
97 \( 1 + 0.474T + 97T^{2} \)
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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

−7.61123267544090142448439128574, −7.29264675704442791149880104779, −6.53500615774141241511320513193, −5.89129647842586634212853860235, −4.81170452713457837776414088006, −4.35492865404284588760414426758, −3.72676501159648313249508079585, −2.78075713017077747452494213559, −1.70794587150615257709954379124, −0.17797704720568139974971665194, 0.17797704720568139974971665194, 1.70794587150615257709954379124, 2.78075713017077747452494213559, 3.72676501159648313249508079585, 4.35492865404284588760414426758, 4.81170452713457837776414088006, 5.89129647842586634212853860235, 6.53500615774141241511320513193, 7.29264675704442791149880104779, 7.61123267544090142448439128574

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