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Properties

Label 2-192-24.5-c0-0-0
Degree $2$
Conductor $192$
Sign $0.707 - 0.707i$
Analytic cond. $0.0958204$
Root an. cond. $0.309548$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 9-s − 2i·19-s − 25-s − i·27-s + 2i·43-s − 49-s + 2·57-s + 2i·67-s + 2·73-s − i·75-s + 81-s − 2·97-s + ⋯
L(s)  = 1  + i·3-s − 9-s − 2i·19-s − 25-s − i·27-s + 2i·43-s − 49-s + 2·57-s + 2i·67-s + 2·73-s − i·75-s + 81-s − 2·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.0958204\)
Root analytic conductor: \(0.309548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6710956517\)
\(L(\frac12)\) \(\approx\) \(0.6710956517\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−12.97722363016898008735771969506, −11.59840981927542118533216491975, −11.01370503130495906728019173891, −9.849516072503568628474637697775, −9.126967973020062447789672133728, −8.018139637964617364667919671293, −6.58586934399064214388814283248, −5.28239110236326378801837962126, −4.24790763788515601708158344657, −2.81568032663380696799101060580, 1.90019379025942779197305225894, 3.62954125332872921564855330771, 5.47366894557275997654851625383, 6.43322565428916493073505123627, 7.63017294754426849302037950971, 8.369648201467567148737270093994, 9.666114570195211975042793979673, 10.82222367714644356398020955651, 11.96652054339395058620004956477, 12.50641606759299101579290172988

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