Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.546 + 0.837i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.56·3-s + 5.73i·5-s + 2.64i·7-s + 11.8·9-s + 1.40·11-s − 19.0i·13-s − 26.1i·15-s − 32.2·17-s − 12.5·19-s − 12.0i·21-s − 15.8i·23-s − 7.86·25-s − 13.0·27-s − 3.29i·29-s − 22.6i·31-s + ⋯
L(s)  = 1  − 1.52·3-s + 1.14i·5-s + 0.377i·7-s + 1.31·9-s + 0.127·11-s − 1.46i·13-s − 1.74i·15-s − 1.89·17-s − 0.661·19-s − 0.575i·21-s − 0.690i·23-s − 0.314·25-s − 0.484·27-s − 0.113i·29-s − 0.731i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.546 + 0.837i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (15, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.546 + 0.837i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0926210 - 0.171085i\)
\(L(\frac12)\)  \(\approx\)  \(0.0926210 - 0.171085i\)
\(L(2)\)   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 \( 1 - 2.64iT \)
good3 \( 1 + 4.56T + 9T^{2} \)
5 \( 1 - 5.73iT - 25T^{2} \)
11 \( 1 - 1.40T + 121T^{2} \)
13 \( 1 + 19.0iT - 169T^{2} \)
17 \( 1 + 32.2T + 289T^{2} \)
19 \( 1 + 12.5T + 361T^{2} \)
23 \( 1 + 15.8iT - 529T^{2} \)
29 \( 1 + 3.29iT - 841T^{2} \)
31 \( 1 + 22.6iT - 961T^{2} \)
37 \( 1 + 54.1iT - 1.36e3T^{2} \)
41 \( 1 + 7.59T + 1.68e3T^{2} \)
43 \( 1 - 20.8T + 1.84e3T^{2} \)
47 \( 1 + 21.6iT - 2.20e3T^{2} \)
53 \( 1 - 0.356iT - 2.80e3T^{2} \)
59 \( 1 + 26.8T + 3.48e3T^{2} \)
61 \( 1 - 86.2iT - 3.72e3T^{2} \)
67 \( 1 + 114.T + 4.48e3T^{2} \)
71 \( 1 - 104. iT - 5.04e3T^{2} \)
73 \( 1 + 24.3T + 5.32e3T^{2} \)
79 \( 1 + 117. iT - 6.24e3T^{2} \)
83 \( 1 + 79.2T + 6.88e3T^{2} \)
89 \( 1 - 2.66T + 7.92e3T^{2} \)
97 \( 1 + 52.0T + 9.40e3T^{2} \)
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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

−11.45733628691509536203140959124, −10.79140068578799233977449563015, −10.30924753755196143245844933768, −8.770700323855977081091246176926, −7.29682983973262612233178177770, −6.39299562918920332854867286191, −5.68512164585197556090120817138, −4.35430646577268367463356695720, −2.57297291506444161396090485057, −0.12962795379553296829796567929, 1.51183478155550317557141775287, 4.36719874430586989441543684022, 4.85660935491023157703044268636, 6.23741470442013812376068450893, 6.94406752730100100576652253844, 8.588748800958123551780917182151, 9.431762252987307922292043061178, 10.74090818118763758149071670301, 11.43211745405745317129325920659, 12.20524020423738395915277380706

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