Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.980 + 0.197i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s i·7-s + (−0.866 + 0.5i)11-s + 0.999i·15-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s + i·27-s + (−0.866 + 0.5i)31-s + (−0.499 + 0.866i)33-s + (0.866 + 0.5i)35-s + (0.5 − 0.866i)37-s + (−0.866 − 0.5i)47-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s i·7-s + (−0.866 + 0.5i)11-s + 0.999i·15-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.5 − 0.866i)21-s + (0.866 + 0.5i)23-s + i·27-s + (−0.866 + 0.5i)31-s + (−0.499 + 0.866i)33-s + (0.866 + 0.5i)35-s + (0.5 − 0.866i)37-s + (−0.866 − 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.980 + 0.197i$
motivic weight  =  \(0\)
character  :  $\chi_{224} (191, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :0),\ 0.980 + 0.197i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.8124955692\)
\(L(\frac12)\)  \(\approx\)  \(0.8124955692\)
\(L(1)\)   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 \( 1 + iT \)
good3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−12.80655480327646462813155737048, −11.18352389090318743678723269802, −10.77165902298516938071181958834, −9.495355798462048788247802205096, −8.294787861306813368008114044733, −7.30376574637874454210651227516, −6.96509250196731627601181733578, −4.97377363559341987547388628072, −3.51637984074493596937248463765, −2.37913706468566554218222193660, 2.50306402926853600546352933742, 3.84728006389649320724127329225, 5.04202855766530028312325826175, 6.27309012268999072264489363596, 8.152010468097277936804154069629, 8.499984918113276897785770765506, 9.309996863911135671874973673591, 10.50908870190490233697565250461, 11.63504597964918013150454296851, 12.66992477175824335103398277134

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