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Properties

Label 2-192-192.107-c1-0-27
Degree $2$
Conductor $192$
Sign $0.465 + 0.884i$
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  + (1.38 + 0.280i)2-s + (−0.682 − 1.59i)3-s + (1.84 + 0.776i)4-s + (−2.10 − 3.15i)5-s + (−0.499 − 2.39i)6-s + (0.358 − 0.866i)7-s + (2.33 + 1.59i)8-s + (−2.06 + 2.17i)9-s + (−2.03 − 4.96i)10-s + (2.16 + 0.431i)11-s + (−0.0204 − 3.46i)12-s + (1.08 + 0.727i)13-s + (0.740 − 1.10i)14-s + (−3.58 + 5.50i)15-s + (2.79 + 2.86i)16-s + (−0.926 − 0.926i)17-s + ⋯
L(s)  = 1  + (0.980 + 0.198i)2-s + (−0.393 − 0.919i)3-s + (0.921 + 0.388i)4-s + (−0.942 − 1.41i)5-s + (−0.203 − 0.978i)6-s + (0.135 − 0.327i)7-s + (0.826 + 0.563i)8-s + (−0.689 + 0.723i)9-s + (−0.644 − 1.56i)10-s + (0.653 + 0.130i)11-s + (−0.00590 − 0.999i)12-s + (0.302 + 0.201i)13-s + (0.197 − 0.294i)14-s + (−0.925 + 1.42i)15-s + (0.698 + 0.715i)16-s + (−0.224 − 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.465 + 0.884i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.465 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42682 - 0.861433i\)
\(L(\frac12)\) \(\approx\) \(1.42682 - 0.861433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.280i)T \)
3 \( 1 + (0.682 + 1.59i)T \)
good5 \( 1 + (2.10 + 3.15i)T + (-1.91 + 4.61i)T^{2} \)
7 \( 1 + (-0.358 + 0.866i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.16 - 0.431i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (-1.08 - 0.727i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (0.926 + 0.926i)T + 17iT^{2} \)
19 \( 1 + (-3.29 - 2.19i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (-2.88 - 6.96i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.54 + 7.79i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + (3.24 + 4.85i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (11.4 - 4.73i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-4.27 - 0.849i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (-5.82 + 5.82i)T - 47iT^{2} \)
53 \( 1 + (1.50 - 7.54i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (10.1 - 6.78i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (1.88 + 9.45i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (3.61 - 0.719i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (5.66 + 2.34i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-7.31 + 3.03i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.775 + 0.775i)T - 79iT^{2} \)
83 \( 1 + (-0.820 + 1.22i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (-7.71 - 3.19i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 - 4.10iT - 97T^{2} \)
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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

−12.23217258921732999173091202163, −11.89427587929163980276022712813, −11.04985383561349965703345714243, −9.086635981655963823479348796293, −7.87299366935237491416044735976, −7.28435564858044009917371291052, −5.87858198687642910459372358071, −4.87082816856896732357000037035, −3.72857028001056925705692366738, −1.43154335852104319294003242788, 2.99279488842949879471300292714, 3.79188420852958133349153741789, 5.03192928895020764237696444080, 6.36158844672310481914480007851, 7.14622494390965802278299171410, 8.788868120273274469717930511015, 10.34379185651465515115413264361, 10.91707182474053933006257384061, 11.61007045136621991948463655548, 12.39400548086128157557167777602

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