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  − 5.85i·3-s − 5.78·5-s + 2.64i·7-s − 25.2·9-s + 3.01i·11-s + 9.78·13-s + 33.8i·15-s − 11.6·17-s − 25.5i·19-s + 15.4·21-s + 26.1i·23-s + 8.42·25-s + 95.4i·27-s + 1.56·29-s − 12.0i·31-s + ⋯
L(s)  = 1  − 1.95i·3-s − 1.15·5-s + 0.377i·7-s − 2.81·9-s + 0.274i·11-s + 0.752·13-s + 2.25i·15-s − 0.682·17-s − 1.34i·19-s + 0.737·21-s + 1.13i·23-s + 0.337·25-s + 3.53i·27-s + 0.0539·29-s − 0.387i·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  =  \((2,\ 224,\ (\ :1),\ -0.707 - 0.707i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.165724 + 0.400094i\)
\(L(\frac12)\)  \(\approx\)  \(0.165724 + 0.400094i\)
\(L(2)\)   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 \( 1 + 5.85iT - 9T^{2} \)
5 \( 1 + 5.78T + 25T^{2} \)
11 \( 1 - 3.01iT - 121T^{2} \)
13 \( 1 - 9.78T + 169T^{2} \)
17 \( 1 + 11.6T + 289T^{2} \)
19 \( 1 + 25.5iT - 361T^{2} \)
23 \( 1 - 26.1iT - 529T^{2} \)
29 \( 1 - 1.56T + 841T^{2} \)
31 \( 1 + 12.0iT - 961T^{2} \)
37 \( 1 + 70.6T + 1.36e3T^{2} \)
41 \( 1 + 49.8T + 1.68e3T^{2} \)
43 \( 1 + 73.2iT - 1.84e3T^{2} \)
47 \( 1 + 44.2iT - 2.20e3T^{2} \)
53 \( 1 + 54.2T + 2.80e3T^{2} \)
59 \( 1 - 12.4iT - 3.48e3T^{2} \)
61 \( 1 - 35.6T + 3.72e3T^{2} \)
67 \( 1 + 24.4iT - 4.48e3T^{2} \)
71 \( 1 - 11.0iT - 5.04e3T^{2} \)
73 \( 1 - 74.3T + 5.32e3T^{2} \)
79 \( 1 + 22.8iT - 6.24e3T^{2} \)
83 \( 1 + 48.1iT - 6.88e3T^{2} \)
89 \( 1 - 67.4T + 7.92e3T^{2} \)
97 \( 1 + 7.75T + 9.40e3T^{2} \)
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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

−11.71181559042772851237990714130, −11.00414167779311780641214216954, −8.925886975818080179900072348699, −8.276241446320759775154473173671, −7.26709286428471378469727880771, −6.67906752233349378235312101423, −5.33306535825978581483637634575, −3.41450533786963782326992307115, −1.92057028481631718850800641211, −0.22975822131839937082282203411, 3.32536055898489011179326849833, 4.01417257710490480911557626573, 4.94385049692587675553021475238, 6.32017512748308473546800351418, 8.116982743974330659796586445834, 8.686873668473789907842523660365, 9.890298197147183689324568885653, 10.74179106710631240128089140198, 11.28741046849696635312037752866, 12.29986700929205627204413473391

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