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  + 0.381·2-s − 1.85·4-s + (0.618 − 1.07i)5-s + (2.11 + 3.66i)7-s − 1.47·8-s + (0.236 − 0.408i)10-s + 3.61·11-s + (−0.690 − 1.19i)13-s + (0.809 + 1.40i)14-s + 3.14·16-s + (3.04 + 5.27i)17-s + (0.618 − 1.07i)19-s + (−1.14 + 1.98i)20-s + 1.38·22-s + (2.19 − 3.79i)23-s + ⋯
L(s)  = 1  + 0.270·2-s − 0.927·4-s + (0.276 − 0.478i)5-s + (0.800 + 1.38i)7-s − 0.520·8-s + (0.0746 − 0.129i)10-s + 1.09·11-s + (−0.191 − 0.331i)13-s + (0.216 + 0.374i)14-s + 0.786·16-s + (0.738 + 1.27i)17-s + (0.141 − 0.245i)19-s + (−0.256 + 0.443i)20-s + 0.294·22-s + (0.456 − 0.791i)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  =  \((2,\ 387,\ (\ :1/2),\ 0.905 - 0.423i)\)
\(L(1)\)  \(\approx\)  \(1.44208 + 0.320486i\)
\(L(\frac12)\)  \(\approx\)  \(1.44208 + 0.320486i\)
\(L(\frac{3}{2})\)   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 \( 1 - 0.381T + 2T^{2} \)
5 \( 1 + (-0.618 + 1.07i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.11 - 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.61T + 11T^{2} \)
13 \( 1 + (0.690 + 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.04 - 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.618 + 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.19 + 3.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.42 + 4.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.47T + 41T^{2} \)
47 \( 1 + 1.14T + 47T^{2} \)
53 \( 1 + (0.690 - 1.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + (1.42 + 2.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.92 + 3.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.39 + 9.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.927 + 1.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.690 - 1.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.01 + 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.927 - 1.60i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.23T + 97T^{2} \)
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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

−11.67424772199226228549382843060, −10.40553894303618823492690906554, −9.238425515540963242494942149328, −8.790394438421281215097388746165, −8.016124011734704576909192541861, −6.31475717651506146810925655415, −5.38609937878047964828311711845, −4.69882702522476547129032413739, −3.33167041264495287755344512937, −1.55474789094885735567747694563, 1.16138409765984000592841574666, 3.31530015010412508927673336024, 4.33339917858814747828843287639, 5.16876031625726942726943532333, 6.58915341542623029717931523263, 7.46024258965996841981214748531, 8.492834043760059689621063638627, 9.644763845377715479679126894531, 10.15894534483485169998372369388, 11.38681074325896555258522418680

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