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Properties

Label 2-3192-3192.2621-c0-0-11
Degree $2$
Conductor $3192$
Sign $-0.954 + 0.296i$
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  + (0.866 − 0.5i)2-s + (−0.342 − 0.939i)3-s + (0.499 − 0.866i)4-s + (−0.766 − 0.642i)6-s + (0.173 − 0.984i)7-s − 0.999i·8-s + (−0.766 + 0.642i)9-s + (−0.984 − 0.173i)12-s − 0.347i·13-s + (−0.342 − 0.939i)14-s + (−0.5 − 0.866i)16-s + (0.939 − 1.62i)17-s + (−0.342 + 0.939i)18-s + (−0.866 + 0.5i)19-s + (−0.984 + 0.173i)21-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.342 − 0.939i)3-s + (0.499 − 0.866i)4-s + (−0.766 − 0.642i)6-s + (0.173 − 0.984i)7-s − 0.999i·8-s + (−0.766 + 0.642i)9-s + (−0.984 − 0.173i)12-s − 0.347i·13-s + (−0.342 − 0.939i)14-s + (−0.5 − 0.866i)16-s + (0.939 − 1.62i)17-s + (−0.342 + 0.939i)18-s + (−0.866 + 0.5i)19-s + (−0.984 + 0.173i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.954 + 0.296i$
Analytic conductor: \(1.59301\)
Root analytic conductor: \(1.26214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3192} (2621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3192,\ (\ :0),\ -0.954 + 0.296i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.767181358\)
\(L(\frac12)\) \(\approx\) \(1.767181358\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.342 + 0.939i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 0.347iT - T^{2} \)
17 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.53iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.62 + 0.939i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.342 + 0.592i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.642 + 1.11i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.254780545097994750962261716935, −7.54627002111252365473592591753, −6.85865342049085588823098523019, −6.34359703023976138771812679321, −5.12563760783831688890445186903, −4.98928676468412727519427301230, −3.60773897327391233123917194239, −2.93921234166794158473293545477, −1.75073290668055382642974212076, −0.845989587367407715093163783675, 2.08906710719566979541769130331, 3.00639047977822814268995809553, 4.03479657514882606983227935792, 4.47572824154237059185446252359, 5.49866311781260380780542466930, 5.99592619726116186175617971009, 6.49496718655254726117190651521, 7.83630977914682557326175293420, 8.311578903065941209930163194144, 9.126404926942034392960838938061

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