Properties

Label 2-192-1.1-c3-0-3
Degree $2$
Conductor $192$
Sign $1$
Analytic cond. $11.3283$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 18·5-s + 8·7-s + 9·9-s − 36·11-s + 10·13-s − 54·15-s + 18·17-s + 100·19-s − 24·21-s + 72·23-s + 199·25-s − 27·27-s + 234·29-s − 16·31-s + 108·33-s + 144·35-s + 226·37-s − 30·39-s + 90·41-s − 452·43-s + 162·45-s + 432·47-s − 279·49-s − 54·51-s − 414·53-s − 648·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.60·5-s + 0.431·7-s + 1/3·9-s − 0.986·11-s + 0.213·13-s − 0.929·15-s + 0.256·17-s + 1.20·19-s − 0.249·21-s + 0.652·23-s + 1.59·25-s − 0.192·27-s + 1.49·29-s − 0.0926·31-s + 0.569·33-s + 0.695·35-s + 1.00·37-s − 0.123·39-s + 0.342·41-s − 1.60·43-s + 0.536·45-s + 1.34·47-s − 0.813·49-s − 0.148·51-s − 1.07·53-s − 1.58·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $1$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{192} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.982246874\)
\(L(\frac12)\) \(\approx\) \(1.982246874\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
good5 \( 1 - 18 T + p^{3} T^{2} \)
7 \( 1 - 8 T + p^{3} T^{2} \)
11 \( 1 + 36 T + p^{3} T^{2} \)
13 \( 1 - 10 T + p^{3} T^{2} \)
17 \( 1 - 18 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 - 72 T + p^{3} T^{2} \)
29 \( 1 - 234 T + p^{3} T^{2} \)
31 \( 1 + 16 T + p^{3} T^{2} \)
37 \( 1 - 226 T + p^{3} T^{2} \)
41 \( 1 - 90 T + p^{3} T^{2} \)
43 \( 1 + 452 T + p^{3} T^{2} \)
47 \( 1 - 432 T + p^{3} T^{2} \)
53 \( 1 + 414 T + p^{3} T^{2} \)
59 \( 1 - 684 T + p^{3} T^{2} \)
61 \( 1 + 422 T + p^{3} T^{2} \)
67 \( 1 + 332 T + p^{3} T^{2} \)
71 \( 1 + 360 T + p^{3} T^{2} \)
73 \( 1 - 26 T + p^{3} T^{2} \)
79 \( 1 - 512 T + p^{3} T^{2} \)
83 \( 1 - 1188 T + p^{3} T^{2} \)
89 \( 1 + 630 T + p^{3} T^{2} \)
97 \( 1 + 1054 T + p^{3} T^{2} \)
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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

−12.10282906437977067245656720674, −10.93496233959272740477103076100, −10.15460424368165076986678220047, −9.348552313724083492806752449448, −7.994183878560034924620737621956, −6.66955231056156118049695270080, −5.61446723029639982983232369317, −4.92178425917919502491900326334, −2.75075683464933758027365201222, −1.25692151493059555444268952686, 1.25692151493059555444268952686, 2.75075683464933758027365201222, 4.92178425917919502491900326334, 5.61446723029639982983232369317, 6.66955231056156118049695270080, 7.994183878560034924620737621956, 9.348552313724083492806752449448, 10.15460424368165076986678220047, 10.93496233959272740477103076100, 12.10282906437977067245656720674

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