Properties

Label 2-192-24.5-c2-0-0
Degree $2$
Conductor $192$
Sign $-0.965 + 0.258i$
Analytic cond. $5.23162$
Root an. cond. $2.28727$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 13.8·7-s − 9·9-s − 13.8i·13-s + 26i·19-s − 41.5i·21-s − 25·25-s − 27i·27-s − 41.5·31-s + 69.2i·37-s + 41.5·39-s + 22i·43-s + 142.·49-s − 78·57-s + 96.9i·61-s + ⋯
L(s)  = 1  + i·3-s − 1.97·7-s − 9-s − 1.06i·13-s + 1.36i·19-s − 1.97i·21-s − 25-s i·27-s − 1.34·31-s + 1.87i·37-s + 1.06·39-s + 0.511i·43-s + 2.91·49-s − 1.36·57-s + 1.59i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(5.23162\)
Root analytic conductor: \(2.28727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1),\ -0.965 + 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0395172 - 0.300163i\)
\(L(\frac12)\) \(\approx\) \(0.0395172 - 0.300163i\)
\(L(2)\) 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 \( 1 - 3iT \)
good5 \( 1 + 25T^{2} \)
7 \( 1 + 13.8T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + 13.8iT - 169T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 26iT - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + 41.5T + 961T^{2} \)
37 \( 1 - 69.2iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 22iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 96.9iT - 3.72e3T^{2} \)
67 \( 1 + 122iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46T + 5.32e3T^{2} \)
79 \( 1 - 69.2T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 2T + 9.40e3T^{2} \)
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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

−12.82806731746393942678352391187, −11.90603511323341784069394192507, −10.48454750273254635050689006313, −9.965254003586122330018809627887, −9.163830152649048538013521294702, −7.88704719590404128589035213691, −6.34257010616382112634496855097, −5.53822672447333158719157056247, −3.83580127306799650174026392483, −3.03334146448950800238454401678, 0.16298038992917204537622835190, 2.35833908791691110885658826204, 3.71258373766346941365330257951, 5.71525080917523646903534025464, 6.69425103238673664280977935965, 7.30696737000570852536065525179, 8.944506841482809357359823466338, 9.518068818003832835972959247380, 10.94849675682141981372177576642, 12.00033641830243601698221242982

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