Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.849 + 0.527i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.44·3-s − 4.88i·5-s + 2.64i·7-s + 2.84·9-s + 21.4·11-s − 13.0i·13-s − 16.8i·15-s − 0.234·17-s − 4.55·19-s + 9.10i·21-s − 10.9i·23-s + 1.15·25-s − 21.1·27-s + 34.6i·29-s + 34.1i·31-s + ⋯
L(s)  = 1  + 1.14·3-s − 0.976i·5-s + 0.377i·7-s + 0.315·9-s + 1.95·11-s − 1.00i·13-s − 1.12i·15-s − 0.0138·17-s − 0.239·19-s + 0.433i·21-s − 0.476i·23-s + 0.0463·25-s − 0.784·27-s + 1.19i·29-s + 1.10i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.849 + 0.527i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (15, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.849 + 0.527i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.22037 - 0.633030i\)
\(L(\frac12)\)  \(\approx\)  \(2.22037 - 0.633030i\)
\(L(2)\)   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 \( 1 - 3.44T + 9T^{2} \)
5 \( 1 + 4.88iT - 25T^{2} \)
11 \( 1 - 21.4T + 121T^{2} \)
13 \( 1 + 13.0iT - 169T^{2} \)
17 \( 1 + 0.234T + 289T^{2} \)
19 \( 1 + 4.55T + 361T^{2} \)
23 \( 1 + 10.9iT - 529T^{2} \)
29 \( 1 - 34.6iT - 841T^{2} \)
31 \( 1 - 34.1iT - 961T^{2} \)
37 \( 1 + 54.2iT - 1.36e3T^{2} \)
41 \( 1 + 37.8T + 1.68e3T^{2} \)
43 \( 1 - 4.84T + 1.84e3T^{2} \)
47 \( 1 - 72.3iT - 2.20e3T^{2} \)
53 \( 1 - 21.6iT - 2.80e3T^{2} \)
59 \( 1 + 34.9T + 3.48e3T^{2} \)
61 \( 1 - 63.6iT - 3.72e3T^{2} \)
67 \( 1 + 18.4T + 4.48e3T^{2} \)
71 \( 1 - 47.5iT - 5.04e3T^{2} \)
73 \( 1 - 55.9T + 5.32e3T^{2} \)
79 \( 1 - 95.0iT - 6.24e3T^{2} \)
83 \( 1 + 71.5T + 6.88e3T^{2} \)
89 \( 1 + 159.T + 7.92e3T^{2} \)
97 \( 1 + 90.4T + 9.40e3T^{2} \)
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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

−12.28631947139710421178677186798, −10.99631090702609171665653949240, −9.572049474687624186651949802229, −8.829791167715341292713768965781, −8.411140393171002711243230243926, −7.01848653873810257268238655519, −5.64342608199672892424649595278, −4.26393930904899217452118623932, −3.07344323878881009943488113422, −1.39260911693242028043536374185, 1.90320483923375796969158310344, 3.34567381800936061227059017084, 4.19608644914000489959271259883, 6.33759538875068458910130569351, 7.02202477564100603289171543366, 8.223480446038677627929243751577, 9.217819071727530692067415598329, 9.904390981339291629539643025697, 11.30247782853388715428167856744, 11.86646181683753514385602231441

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