Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.707 + 0.707i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.55i·3-s + 9.86·5-s − 2.64i·7-s + 2.47·9-s + 13.1i·11-s − 5.86·13-s − 25.2i·15-s − 0.570·17-s − 15.6i·19-s − 6.75·21-s + 16.4i·23-s + 72.3·25-s − 29.3i·27-s − 29.7·29-s − 54.8i·31-s + ⋯
L(s)  = 1  − 0.851i·3-s + 1.97·5-s − 0.377i·7-s + 0.274·9-s + 1.19i·11-s − 0.451·13-s − 1.68i·15-s − 0.0335·17-s − 0.824i·19-s − 0.321·21-s + 0.716i·23-s + 2.89·25-s − 1.08i·27-s − 1.02·29-s − 1.76i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.707 + 0.707i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.707 + 0.707i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.97408 - 0.817693i\)
\(L(\frac12)\)  \(\approx\)  \(1.97408 - 0.817693i\)
\(L(2)\)   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 \( 1 + 2.55iT - 9T^{2} \)
5 \( 1 - 9.86T + 25T^{2} \)
11 \( 1 - 13.1iT - 121T^{2} \)
13 \( 1 + 5.86T + 169T^{2} \)
17 \( 1 + 0.570T + 289T^{2} \)
19 \( 1 + 15.6iT - 361T^{2} \)
23 \( 1 - 16.4iT - 529T^{2} \)
29 \( 1 + 29.7T + 841T^{2} \)
31 \( 1 + 54.8iT - 961T^{2} \)
37 \( 1 + 42.0T + 1.36e3T^{2} \)
41 \( 1 - 0.773T + 1.68e3T^{2} \)
43 \( 1 - 41.7iT - 1.84e3T^{2} \)
47 \( 1 - 58.4iT - 2.20e3T^{2} \)
53 \( 1 - 5.65T + 2.80e3T^{2} \)
59 \( 1 - 42.6iT - 3.48e3T^{2} \)
61 \( 1 + 95.9T + 3.72e3T^{2} \)
67 \( 1 - 69.8iT - 4.48e3T^{2} \)
71 \( 1 + 92.0iT - 5.04e3T^{2} \)
73 \( 1 - 9.97T + 5.32e3T^{2} \)
79 \( 1 + 20.1iT - 6.24e3T^{2} \)
83 \( 1 - 151. iT - 6.88e3T^{2} \)
89 \( 1 - 5.79T + 7.92e3T^{2} \)
97 \( 1 - 103.T + 9.40e3T^{2} \)
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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

−12.21126460813623934465187386659, −10.78926818907266438136548721512, −9.739204899573004597039945102407, −9.357300196076271496049307469757, −7.58887242433735179077140025045, −6.83676097967662332829249063265, −5.85835182976470628779714902298, −4.66008621705535079831659915351, −2.40456534071269030521606587998, −1.50098197212991132308958801179, 1.79837948263342960039456953692, 3.29861522692108960745638425725, 5.03800970056913530524778985981, 5.73278127810613671751350318584, 6.80420481128837437157313154216, 8.657437091751355184823086594959, 9.305445802290685809528723453070, 10.28971681132868143795494725762, 10.66995356125795292875394202670, 12.26164882891466705626303154017

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