Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.657 - 0.753i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.94 + 3.37i)3-s + (−4.42 + 7.67i)5-s + (6.92 − 1.03i)7-s + (−3.09 + 5.35i)9-s + (3.15 − 1.82i)11-s − 7.79·13-s − 34.5·15-s + (−9.07 + 5.23i)17-s + (−5.39 + 9.34i)19-s + (16.9 + 21.3i)21-s + (−6.45 + 11.1i)23-s + (−26.7 − 46.3i)25-s + 10.9·27-s − 17.2i·29-s + (26.1 − 15.1i)31-s + ⋯
L(s)  = 1  + (0.649 + 1.12i)3-s + (−0.885 + 1.53i)5-s + (0.989 − 0.147i)7-s + (−0.343 + 0.594i)9-s + (0.287 − 0.165i)11-s − 0.599·13-s − 2.30·15-s + (−0.533 + 0.308i)17-s + (−0.283 + 0.491i)19-s + (0.808 + 1.01i)21-s + (−0.280 + 0.486i)23-s + (−1.06 − 1.85i)25-s + 0.406·27-s − 0.594i·29-s + (0.844 − 0.487i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.657 - 0.753i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (145, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.657 - 0.753i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.685394 + 1.50795i\)
\(L(\frac12)\)  \(\approx\)  \(0.685394 + 1.50795i\)
\(L(2)\)   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 \( 1 + (-6.92 + 1.03i)T \)
good3 \( 1 + (-1.94 - 3.37i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (4.42 - 7.67i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-3.15 + 1.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 7.79T + 169T^{2} \)
17 \( 1 + (9.07 - 5.23i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (5.39 - 9.34i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (6.45 - 11.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 17.2iT - 841T^{2} \)
31 \( 1 + (-26.1 + 15.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-34.2 - 19.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 73.6iT - 1.68e3T^{2} \)
43 \( 1 + 40.8iT - 1.84e3T^{2} \)
47 \( 1 + (-36.2 - 20.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (5.55 - 3.20i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-7.95 - 13.7i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (6.07 - 10.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (6.75 - 3.89i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 41.3T + 5.04e3T^{2} \)
73 \( 1 + (-77.6 + 44.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-35.3 + 61.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 60.8T + 6.88e3T^{2} \)
89 \( 1 + (23.4 + 13.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 3.26iT - 9.40e3T^{2} \)
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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

−11.97693549299699794557168384871, −11.22356807371583018159381018337, −10.47821745352505864976345231075, −9.653686078977874116872648616657, −8.312098468557966139808422884776, −7.61281743312980869417888657292, −6.35662765581023485373847832234, −4.54334041450688068671045349957, −3.78650174445159625533375802946, −2.59407949589186077315603512911, 0.888892850848509171333454312569, 2.22296676223206936382118416601, 4.26787163026277701201008303932, 5.14176450883272572540563182866, 6.94578856257837151606083756442, 7.87531999231234154199544547201, 8.478949526925327871811557211051, 9.230645351090332748385058292202, 11.01347397798080996784429047701, 12.13445738266258683450997916150

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