Properties

Degree 2
Conductor $ 2 \cdot 7 $
Sign $0.874 + 0.485i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (0.5 + 0.866i)3-s + (−1.99 − 3.46i)4-s + (−3.5 + 6.06i)5-s + 1.99·6-s + (−10 + 15.5i)7-s − 7.99·8-s + (13 − 22.5i)9-s + (7 + 12.1i)10-s + (−17.5 − 30.3i)11-s + (1.99 − 3.46i)12-s + 66·13-s + (17 + 32.9i)14-s − 7·15-s + (−8 + 13.8i)16-s + (−29.5 − 51.0i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.0962 + 0.166i)3-s + (−0.249 − 0.433i)4-s + (−0.313 + 0.542i)5-s + 0.136·6-s + (−0.539 + 0.841i)7-s − 0.353·8-s + (0.481 − 0.833i)9-s + (0.221 + 0.383i)10-s + (−0.479 − 0.830i)11-s + (0.0481 − 0.0833i)12-s + 1.40·13-s + (0.324 + 0.628i)14-s − 0.120·15-s + (−0.125 + 0.216i)16-s + (−0.420 − 0.728i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(14\)    =    \(2 \cdot 7\)
\( \varepsilon \)  =  $0.874 + 0.485i$
motivic weight  =  \(3\)
character  :  $\chi_{14} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 14,\ (\ :3/2),\ 0.874 + 0.485i)\)
\(L(2)\)  \(\approx\)  \(1.03971 - 0.269364i\)
\(L(\frac12)\)  \(\approx\)  \(1.03971 - 0.269364i\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
7 \( 1 + (10 - 15.5i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (3.5 - 6.06i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (17.5 + 30.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 66T + 2.19e3T^{2} \)
17 \( 1 + (29.5 + 51.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (68.5 - 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 106T + 2.43e4T^{2} \)
31 \( 1 + (37.5 + 64.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 498T + 6.89e4T^{2} \)
43 \( 1 - 260T + 7.95e4T^{2} \)
47 \( 1 + (-85.5 + 148. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-208.5 - 361. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-8.5 - 14.7i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (25.5 - 44.1i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (219.5 + 380. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 784T + 3.57e5T^{2} \)
73 \( 1 + (147.5 + 255. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-247.5 + 428. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 932T + 5.71e5T^{2} \)
89 \( 1 + (-436.5 + 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 290T + 9.12e5T^{2} \)
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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

−18.84861829834191611688641591318, −18.42395395886323971363820162841, −16.03861877898491807075026006561, −15.00722107260579078507327498172, −13.38296499294512487966267915579, −11.97993937386865071289635495549, −10.54641437726097788540688220612, −8.839065203126092887000071158281, −6.17419372141557392180812905306, −3.44148627176067932895503895911, 4.45809912835375765151201974082, 6.86227351184222756791851788691, 8.452930461734506816688691865851, 10.60580380214991759514489550893, 12.82745157815110757261047381609, 13.54746730171127226533518152529, 15.47552069754499605822576797552, 16.38042466698445010529767813229, 17.73376855449346056640164989882, 19.36659132660275986647232450062

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