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Properties

Label 2-192-16.13-c1-0-3
Degree $2$
Conductor $192$
Sign $0.513 + 0.857i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)3-s + (1.27 − 1.27i)5-s − 0.158i·7-s + 1.00i·9-s + (3.79 − 3.79i)11-s + (−4.21 − 4.21i)13-s − 1.79·15-s + 3.05·17-s + (2.15 + 2.15i)19-s + (−0.112 + 0.112i)21-s + 2.82i·23-s + 1.76i·25-s + (0.707 − 0.707i)27-s + (2.09 + 2.09i)29-s − 4.15·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.568 − 0.568i)5-s − 0.0600i·7-s + 0.333i·9-s + (1.14 − 1.14i)11-s + (−1.16 − 1.16i)13-s − 0.464·15-s + 0.740·17-s + (0.495 + 0.495i)19-s + (−0.0245 + 0.0245i)21-s + 0.589i·23-s + 0.353i·25-s + (0.136 − 0.136i)27-s + (0.389 + 0.389i)29-s − 0.746·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.513 + 0.857i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.513 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.988890 - 0.560387i\)
\(L(\frac12)\) \(\approx\) \(0.988890 - 0.560387i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (-1.27 + 1.27i)T - 5iT^{2} \)
7 \( 1 + 0.158iT - 7T^{2} \)
11 \( 1 + (-3.79 + 3.79i)T - 11iT^{2} \)
13 \( 1 + (4.21 + 4.21i)T + 13iT^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
19 \( 1 + (-2.15 - 2.15i)T + 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (-2.09 - 2.09i)T + 29iT^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 + (5.98 - 5.98i)T - 37iT^{2} \)
41 \( 1 - 2.60iT - 41T^{2} \)
43 \( 1 + (5.75 - 5.75i)T - 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-3.55 + 3.55i)T - 53iT^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 + (-3.66 - 3.66i)T + 61iT^{2} \)
67 \( 1 + (0.767 + 0.767i)T + 67iT^{2} \)
71 \( 1 - 0.317iT - 71T^{2} \)
73 \( 1 - 1.33iT - 73T^{2} \)
79 \( 1 - 9.69T + 79T^{2} \)
83 \( 1 + (0.115 + 0.115i)T + 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + 0.571T + 97T^{2} \)
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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

−12.32427035096230560803436669269, −11.62323711111200958754092101823, −10.36889773308737704783543982604, −9.447744822486151092377390517691, −8.322741191113076581470058029429, −7.20898993143409779916004574814, −5.88520125345707706699712368694, −5.17148727733757249257006093125, −3.30929555236875743010133731146, −1.24264254162198050576480100326, 2.17333486573467991237480334359, 4.03125249694446395002958160784, 5.18640800857765897411880961294, 6.58402442750176547034362222669, 7.23679079770375296003894875878, 9.099073856968484862542039367852, 9.746443733095670553139433762766, 10.60063098243362821913589421191, 11.92860753412365095676112117674, 12.28394846118638302003926688258

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