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Properties

Label 2-3192-3192.2309-c0-0-2
Degree $2$
Conductor $3192$
Sign $-0.660 - 0.750i$
Analytic cond. $1.59301$
Root an. cond. $1.26214$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)2-s + (0.984 − 0.173i)3-s + (−0.939 + 0.342i)4-s + (0.118 − 0.326i)5-s + (0.342 + 0.939i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.939 − 0.342i)9-s + (0.342 + 0.0603i)10-s + (−0.866 + 0.5i)12-s + (−1.20 + 1.43i)13-s + (−0.939 − 0.342i)14-s + (0.0603 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.5 + 0.866i)18-s + (−0.984 + 0.173i)19-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.984 − 0.173i)3-s + (−0.939 + 0.342i)4-s + (0.118 − 0.326i)5-s + (0.342 + 0.939i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.939 − 0.342i)9-s + (0.342 + 0.0603i)10-s + (−0.866 + 0.5i)12-s + (−1.20 + 1.43i)13-s + (−0.939 − 0.342i)14-s + (0.0603 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.5 + 0.866i)18-s + (−0.984 + 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.660 - 0.750i$
Analytic conductor: \(1.59301\)
Root analytic conductor: \(1.26214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3192} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3192,\ (\ :0),\ -0.660 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.472629763\)
\(L(\frac12)\) \(\approx\) \(1.472629763\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.984 - 0.173i)T \)
good5 \( 1 + (-0.118 + 0.326i)T + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.20 - 1.43i)T + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.223 - 1.26i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-1.20 + 0.439i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (1.32 + 0.766i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.939 + 0.342i)T^{2} \)
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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

−8.848031449440135108281484647592, −8.593330833242536752487287109614, −7.41472446371091251969584677248, −7.08222427936977934410203039327, −6.25494364046094153159548149822, −5.32399920924301241775982574473, −4.57894340193634725151414673070, −3.71677860404665465888075560897, −2.78026673657496065111108622509, −1.72328470186867443804892806328, 0.74598011841606071217000921999, 2.37853251376917190635343652448, 2.77369172208854216297896247568, 3.65518345923739576531744128793, 4.49028613397443116777618147695, 5.10836269818787684945121246636, 6.44218675897300695100281393218, 7.15244902828603055090120967831, 8.131917916663559748983320903394, 8.589406089571826328034013703402

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