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Properties

Label 2-192-48.35-c1-0-4
Degree $2$
Conductor $192$
Sign $0.557 + 0.830i$
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  + (1.52 − 0.814i)3-s + (−2.08 − 2.08i)5-s + 1.14·7-s + (1.67 − 2.48i)9-s + (1.67 − 1.67i)11-s + (0.146 + 0.146i)13-s + (−4.88 − 1.48i)15-s + 5.59i·17-s + (−1.48 + 1.48i)19-s + (1.75 − 0.933i)21-s + 3.34i·23-s + 3.68i·25-s + (0.533 − 5.16i)27-s + (3.51 − 3.51i)29-s + 5.83i·31-s + ⋯
L(s)  = 1  + (0.882 − 0.470i)3-s + (−0.931 − 0.931i)5-s + 0.433·7-s + (0.558 − 0.829i)9-s + (0.504 − 0.504i)11-s + (0.0405 + 0.0405i)13-s + (−1.26 − 0.384i)15-s + 1.35i·17-s + (−0.341 + 0.341i)19-s + (0.382 − 0.203i)21-s + 0.698i·23-s + 0.737i·25-s + (0.102 − 0.994i)27-s + (0.652 − 0.652i)29-s + 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.557 + 0.830i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24104 - 0.661499i\)
\(L(\frac12)\) \(\approx\) \(1.24104 - 0.661499i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 0.814i)T \)
good5 \( 1 + (2.08 + 2.08i)T + 5iT^{2} \)
7 \( 1 - 1.14T + 7T^{2} \)
11 \( 1 + (-1.67 + 1.67i)T - 11iT^{2} \)
13 \( 1 + (-0.146 - 0.146i)T + 13iT^{2} \)
17 \( 1 - 5.59iT - 17T^{2} \)
19 \( 1 + (1.48 - 1.48i)T - 19iT^{2} \)
23 \( 1 - 3.34iT - 23T^{2} \)
29 \( 1 + (-3.51 + 3.51i)T - 29iT^{2} \)
31 \( 1 - 5.83iT - 31T^{2} \)
37 \( 1 + (4.83 - 4.83i)T - 37iT^{2} \)
41 \( 1 + 0.610T + 41T^{2} \)
43 \( 1 + (-1.48 - 1.48i)T + 43iT^{2} \)
47 \( 1 - 6.41T + 47T^{2} \)
53 \( 1 + (0.164 + 0.164i)T + 53iT^{2} \)
59 \( 1 + (9.05 - 9.05i)T - 59iT^{2} \)
61 \( 1 + (-4.53 - 4.53i)T + 61iT^{2} \)
67 \( 1 + (-0.635 + 0.635i)T - 67iT^{2} \)
71 \( 1 + 6.90iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 + 9.83iT - 79T^{2} \)
83 \( 1 + (8.09 + 8.09i)T + 83iT^{2} \)
89 \( 1 + 0.490T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−12.35916770193432408241206837790, −11.76037685201257836332262358627, −10.38783285238334889190897572522, −8.936077931240617284171765625879, −8.403932312579022152713590721226, −7.60671612098800409429148685735, −6.21501490472414589172221538847, −4.50986112245900434132508594839, −3.49887565166340526902831036167, −1.44832875490441251134371734677, 2.53507531515238817052259123893, 3.77225188269583600126601590197, 4.83752565753354598899986895864, 6.86275708974916046506566892917, 7.59499439674379098439458430655, 8.654153514659764088642493373203, 9.692197617668385146423089573496, 10.78129245546863075660357510695, 11.51886748429569987747208597004, 12.66977328572611817056940592825

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