Properties

Label 2-192-1.1-c7-0-0
Degree $2$
Conductor $192$
Sign $1$
Analytic cond. $59.9779$
Root an. cond. $7.74454$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27·3-s − 335.·5-s − 710.·7-s + 729·9-s − 2.89e3·11-s − 1.01e4·13-s + 9.05e3·15-s − 1.42e4·17-s − 3.31e4·19-s + 1.91e4·21-s − 7.53e4·23-s + 3.42e4·25-s − 1.96e4·27-s − 1.50e5·29-s + 5.51e4·31-s + 7.80e4·33-s + 2.38e5·35-s − 5.10e5·37-s + 2.73e5·39-s + 6.04e5·41-s + 4.79e5·43-s − 2.44e5·45-s + 1.59e5·47-s − 3.18e5·49-s + 3.84e5·51-s − 3.19e5·53-s + 9.69e5·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.19·5-s − 0.782·7-s + 0.333·9-s − 0.655·11-s − 1.27·13-s + 0.692·15-s − 0.703·17-s − 1.11·19-s + 0.451·21-s − 1.29·23-s + 0.438·25-s − 0.192·27-s − 1.14·29-s + 0.332·31-s + 0.378·33-s + 0.938·35-s − 1.65·37-s + 0.738·39-s + 1.37·41-s + 0.920·43-s − 0.399·45-s + 0.223·47-s − 0.387·49-s + 0.405·51-s − 0.294·53-s + 0.785·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $1$
Analytic conductor: \(59.9779\)
Root analytic conductor: \(7.74454\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.01299044733\)
\(L(\frac12)\) \(\approx\) \(0.01299044733\)
\(L(\frac{9}{2})\) 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 \( 1 + 27T \)
good5 \( 1 + 335.T + 7.81e4T^{2} \)
7 \( 1 + 710.T + 8.23e5T^{2} \)
11 \( 1 + 2.89e3T + 1.94e7T^{2} \)
13 \( 1 + 1.01e4T + 6.27e7T^{2} \)
17 \( 1 + 1.42e4T + 4.10e8T^{2} \)
19 \( 1 + 3.31e4T + 8.93e8T^{2} \)
23 \( 1 + 7.53e4T + 3.40e9T^{2} \)
29 \( 1 + 1.50e5T + 1.72e10T^{2} \)
31 \( 1 - 5.51e4T + 2.75e10T^{2} \)
37 \( 1 + 5.10e5T + 9.49e10T^{2} \)
41 \( 1 - 6.04e5T + 1.94e11T^{2} \)
43 \( 1 - 4.79e5T + 2.71e11T^{2} \)
47 \( 1 - 1.59e5T + 5.06e11T^{2} \)
53 \( 1 + 3.19e5T + 1.17e12T^{2} \)
59 \( 1 + 1.81e6T + 2.48e12T^{2} \)
61 \( 1 + 3.00e6T + 3.14e12T^{2} \)
67 \( 1 - 3.12e6T + 6.06e12T^{2} \)
71 \( 1 - 4.15e6T + 9.09e12T^{2} \)
73 \( 1 - 1.69e6T + 1.10e13T^{2} \)
79 \( 1 + 4.26e6T + 1.92e13T^{2} \)
83 \( 1 - 7.78e6T + 2.71e13T^{2} \)
89 \( 1 + 9.58e6T + 4.42e13T^{2} \)
97 \( 1 + 1.01e7T + 8.07e13T^{2} \)
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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

−11.26040570804327509994363585302, −10.42022227237941806850898957050, −9.354645946453776336684718821393, −8.009741163420491602537835968433, −7.20379990572559263344193492047, −6.09128434337858793643556896668, −4.74167298248039190523714896813, −3.77238977930306924340965385370, −2.29434310319989870698830112531, −0.05851877363439634932366719804, 0.05851877363439634932366719804, 2.29434310319989870698830112531, 3.77238977930306924340965385370, 4.74167298248039190523714896813, 6.09128434337858793643556896668, 7.20379990572559263344193492047, 8.009741163420491602537835968433, 9.354645946453776336684718821393, 10.42022227237941806850898957050, 11.26040570804327509994363585302

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