Properties

Degree 2
Conductor $ 3^{2} \cdot 43 $
Sign $0.905 - 0.423i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·2-s + 4.85·4-s + (−1.61 + 2.80i)5-s + (−0.118 − 0.204i)7-s + 7.47·8-s + (−4.23 + 7.33i)10-s + 1.38·11-s + (−1.80 − 3.13i)13-s + (−0.309 − 0.535i)14-s + 9.85·16-s + (−2.54 − 4.40i)17-s + (−1.61 + 2.80i)19-s + (−7.85 + 13.6i)20-s + 3.61·22-s + (3.30 − 5.73i)23-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.42·4-s + (−0.723 + 1.25i)5-s + (−0.0446 − 0.0772i)7-s + 2.64·8-s + (−1.33 + 2.32i)10-s + 0.416·11-s + (−0.501 − 0.869i)13-s + (−0.0825 − 0.143i)14-s + 2.46·16-s + (−0.617 − 1.06i)17-s + (−0.371 + 0.642i)19-s + (−1.75 + 3.04i)20-s + 0.771·22-s + (0.689 − 1.19i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(387\)    =    \(3^{2} \cdot 43\)
\( \varepsilon \)  =  $0.905 - 0.423i$
motivic weight  =  \(1\)
character  :  $\chi_{387} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 387,\ (\ :1/2),\ 0.905 - 0.423i)\)
\(L(1)\)  \(\approx\)  \(3.38805 + 0.752957i\)
\(L(\frac12)\)  \(\approx\)  \(3.38805 + 0.752957i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;43\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + (6.5 - 0.866i)T \)
good2 \( 1 - 2.61T + 2T^{2} \)
5 \( 1 + (1.61 - 2.80i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.118 + 0.204i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + (1.80 + 3.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.54 + 4.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.61 - 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.30 + 5.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.927 - 1.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.527T + 41T^{2} \)
47 \( 1 + 7.85T + 47T^{2} \)
53 \( 1 + (1.80 - 3.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.09T + 59T^{2} \)
61 \( 1 + (-1.92 - 3.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.42 - 2.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.89 - 11.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.42 - 4.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.80 - 3.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.51 - 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.42 + 4.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.76T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.55292437840667668480482209505, −10.92287633043037631902236927124, −10.08947589812656564809199197520, −8.216962017218923593468545866903, −6.93512064305874472716357360859, −6.77415307266928779591835410604, −5.39760987428249841793093406373, −4.36408407197613184849446283523, −3.31863811460991859938183471188, −2.56327692664283757678581075940, 1.85514563682022815991985534519, 3.53333195714306311824259795575, 4.43861141019056810970362673115, 5.02367580718238670301237165275, 6.24092857516818876140666146577, 7.14967501520449008266804408669, 8.326413393171238875174215777658, 9.388219390513786409918144559829, 10.95001334384411482794288039466, 11.68979292233585407633965676577

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