Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.0298 + 0.999i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.455 + 0.788i)3-s + (3.17 − 5.49i)5-s + (−3.79 − 5.88i)7-s + (4.08 − 7.07i)9-s + (−11.4 + 6.60i)11-s − 19.4·13-s + 5.77·15-s + (13.7 − 7.96i)17-s + (8.22 − 14.2i)19-s + (2.91 − 5.67i)21-s + (11.9 − 20.7i)23-s + (−7.62 − 13.2i)25-s + 15.6·27-s + 16.6i·29-s + (11.1 − 6.42i)31-s + ⋯
L(s)  = 1  + (0.151 + 0.262i)3-s + (0.634 − 1.09i)5-s + (−0.541 − 0.840i)7-s + (0.453 − 0.786i)9-s + (−1.04 + 0.600i)11-s − 1.49·13-s + 0.385·15-s + (0.811 − 0.468i)17-s + (0.433 − 0.750i)19-s + (0.138 − 0.270i)21-s + (0.520 − 0.900i)23-s + (−0.305 − 0.528i)25-s + 0.579·27-s + 0.574i·29-s + (0.359 − 0.207i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0298 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0298 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.0298 + 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (145, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.0298 + 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.04414 - 1.01338i\)
\(L(\frac12)\)  \(\approx\)  \(1.04414 - 1.01338i\)
\(L(2)\)   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 \( 1 + (3.79 + 5.88i)T \)
good3 \( 1 + (-0.455 - 0.788i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-3.17 + 5.49i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (11.4 - 6.60i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 19.4T + 169T^{2} \)
17 \( 1 + (-13.7 + 7.96i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.22 + 14.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-11.9 + 20.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 16.6iT - 841T^{2} \)
31 \( 1 + (-11.1 + 6.42i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-41.1 - 23.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 6.49iT - 1.68e3T^{2} \)
43 \( 1 + 33.2iT - 1.84e3T^{2} \)
47 \( 1 + (-18.9 - 10.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (32.2 - 18.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-27.3 - 47.3i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (5.12 - 8.87i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-14.8 + 8.56i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 32.0T + 5.04e3T^{2} \)
73 \( 1 + (-92.8 + 53.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (29.1 - 50.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 36.3T + 6.88e3T^{2} \)
89 \( 1 + (-0.929 - 0.536i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 169. iT - 9.40e3T^{2} \)
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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

−12.14527627262231697616838901692, −10.46282380483697743204339272556, −9.738180342405858635594790436504, −9.203287215888444148765481747357, −7.70666429908579976649914229058, −6.81626177311665151472437352553, −5.22603479128425245397216840599, −4.49376633471657431179534414803, −2.79220255199821926926971028266, −0.77854643502638731759591847151, 2.24675742952826756229698691915, 3.08095627333425054579685747770, 5.19085273232258184951949937082, 6.06637522025694131180855187085, 7.31635718012869175560170009916, 8.067667869187045806594691340087, 9.704227982354203163890679198850, 10.12906039375478206960023396092, 11.20972036490551731472420022993, 12.42204896127312699778682374225

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