Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.670 - 0.742i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.43 − 0.830i)3-s + (7.27 + 4.20i)5-s + (−3.99 + 5.74i)7-s + (−3.12 + 5.40i)9-s + (2.60 + 4.51i)11-s + 4.88i·13-s + 13.9·15-s + (6.68 − 3.86i)17-s + (−30.6 − 17.6i)19-s + (−0.979 + 11.5i)21-s + (13.3 − 23.0i)23-s + (22.7 + 39.4i)25-s + 25.3i·27-s + 45.1·29-s + (35.0 − 20.2i)31-s + ⋯
L(s)  = 1  + (0.479 − 0.276i)3-s + (1.45 + 0.840i)5-s + (−0.571 + 0.820i)7-s + (−0.346 + 0.600i)9-s + (0.237 + 0.410i)11-s + 0.375i·13-s + 0.930·15-s + (0.393 − 0.227i)17-s + (−1.61 − 0.929i)19-s + (−0.0466 + 0.551i)21-s + (0.578 − 1.00i)23-s + (0.911 + 1.57i)25-s + 0.937i·27-s + 1.55·29-s + (1.12 − 0.652i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.670 - 0.742i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (33, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.670 - 0.742i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.88464 + 0.837197i\)
\(L(\frac12)\)  \(\approx\)  \(1.88464 + 0.837197i\)
\(L(2)\)   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 \( 1 \)
7 \( 1 + (3.99 - 5.74i)T \)
good3 \( 1 + (-1.43 + 0.830i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-7.27 - 4.20i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.60 - 4.51i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 4.88iT - 169T^{2} \)
17 \( 1 + (-6.68 + 3.86i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (30.6 + 17.6i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-13.3 + 23.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 45.1T + 841T^{2} \)
31 \( 1 + (-35.0 + 20.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (3.97 - 6.89i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 26.6iT - 1.68e3T^{2} \)
43 \( 1 + 0.403T + 1.84e3T^{2} \)
47 \( 1 + (-28.2 - 16.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (40.6 + 70.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-7.62 + 4.40i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (25.3 + 14.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (43.7 + 75.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 27.5T + 5.04e3T^{2} \)
73 \( 1 + (-75.3 + 43.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (60.8 - 105. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 46.6iT - 6.88e3T^{2} \)
89 \( 1 + (52.7 + 30.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 66.4iT - 9.40e3T^{2} \)
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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

−12.36477675866541389186444505002, −11.00139233490932593547246953600, −10.15287150737460828641342902387, −9.236785295816128068938929416410, −8.393940623000535493109092374209, −6.76912953227780992862291015228, −6.26887908752040533816768047370, −4.91998586420649805978407432667, −2.76887458825470767910026283811, −2.20808274445534665079774031857, 1.17782187939062965665763266610, 2.98691146840457627844815349859, 4.34407255049197663843061347853, 5.80789857435806710445992613040, 6.52144545153624042412020396127, 8.252503615418018998477747865186, 9.030636444049728312698726297934, 9.931845155682700898914075678965, 10.51356037534314392354462155829, 12.16789075819051399579397628798

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