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Properties

Label 2-192-192.35-c1-0-8
Degree $2$
Conductor $192$
Sign $0.212 - 0.977i$
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  + (1.35 + 0.411i)2-s + (−0.525 + 1.65i)3-s + (1.66 + 1.11i)4-s + (−0.731 − 0.488i)5-s + (−1.39 + 2.01i)6-s + (0.683 + 1.64i)7-s + (1.78 + 2.18i)8-s + (−2.44 − 1.73i)9-s + (−0.788 − 0.962i)10-s + (−0.385 − 1.93i)11-s + (−2.71 + 2.15i)12-s + (−0.659 − 0.987i)13-s + (0.245 + 2.51i)14-s + (1.19 − 0.950i)15-s + (1.52 + 3.69i)16-s + (2.96 − 2.96i)17-s + ⋯
L(s)  = 1  + (0.956 + 0.290i)2-s + (−0.303 + 0.952i)3-s + (0.830 + 0.556i)4-s + (−0.327 − 0.218i)5-s + (−0.567 + 0.823i)6-s + (0.258 + 0.623i)7-s + (0.632 + 0.774i)8-s + (−0.815 − 0.578i)9-s + (−0.249 − 0.304i)10-s + (−0.116 − 0.584i)11-s + (−0.782 + 0.622i)12-s + (−0.182 − 0.273i)13-s + (0.0656 + 0.671i)14-s + (0.307 − 0.245i)15-s + (0.380 + 0.924i)16-s + (0.720 − 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.212 - 0.977i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.212 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38403 + 1.11530i\)
\(L(\frac12)\) \(\approx\) \(1.38403 + 1.11530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.411i)T \)
3 \( 1 + (0.525 - 1.65i)T \)
good5 \( 1 + (0.731 + 0.488i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (-0.683 - 1.64i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.385 + 1.93i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (0.659 + 0.987i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (-2.96 + 2.96i)T - 17iT^{2} \)
19 \( 1 + (-2.88 - 4.31i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (-1.53 + 3.71i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (8.74 + 1.74i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 - 2.66T + 31T^{2} \)
37 \( 1 + (2.69 + 1.80i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-9.64 - 3.99i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.964 + 4.84i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (2.39 + 2.39i)T + 47iT^{2} \)
53 \( 1 + (13.1 - 2.61i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (5.24 - 7.85i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (-4.01 - 0.798i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (1.21 - 6.11i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (8.97 - 3.71i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.96 + 1.64i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.91 - 3.91i)T + 79iT^{2} \)
83 \( 1 + (7.85 - 5.24i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (7.76 - 3.21i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 2.84iT - 97T^{2} \)
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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

−12.55969163830318929639584440968, −11.84642265464783532875639654053, −11.09281724592108817090894195386, −9.936701078981237192982316944061, −8.632545381269281331067380566067, −7.62516748023709255911692925421, −5.96960424698347889769995729644, −5.31328427053857129430279088866, −4.13544100316387095367097652432, −2.92874617328694207738429044565, 1.60657867792518832027480501315, 3.29386684505147007562860283262, 4.79292648819474195595153132913, 5.93151487159211854543467006360, 7.22590463687760157124015735762, 7.62043767378813990970797025508, 9.563867903959197088825095264884, 10.94577341744248916520570610100, 11.41433675303655014348922745299, 12.47759049770158444318142842436

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