Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.664 - 0.746i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.438 − 0.253i)3-s + (−4.59 − 2.65i)5-s + (−5.27 − 4.59i)7-s + (−4.37 + 7.57i)9-s + (8.54 + 14.8i)11-s + 21.4i·13-s − 2.68·15-s + (−20.7 + 12.0i)17-s + (−10.5 − 6.11i)19-s + (−3.48 − 0.680i)21-s + (20.1 − 34.8i)23-s + (1.55 + 2.69i)25-s + 8.99i·27-s − 26.0·29-s + (−21.8 + 12.6i)31-s + ⋯
L(s)  = 1  + (0.146 − 0.0844i)3-s + (−0.918 − 0.530i)5-s + (−0.753 − 0.656i)7-s + (−0.485 + 0.841i)9-s + (0.776 + 1.34i)11-s + 1.65i·13-s − 0.179·15-s + (−1.22 + 0.706i)17-s + (−0.557 − 0.321i)19-s + (−0.165 − 0.0324i)21-s + (0.875 − 1.51i)23-s + (0.0623 + 0.107i)25-s + 0.333i·27-s − 0.899·29-s + (−0.705 + 0.407i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.664 - 0.746i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (33, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.664 - 0.746i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.191524 + 0.426878i\)
\(L(\frac12)\)  \(\approx\)  \(0.191524 + 0.426878i\)
\(L(2)\)   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 \( 1 + (5.27 + 4.59i)T \)
good3 \( 1 + (-0.438 + 0.253i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (4.59 + 2.65i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-8.54 - 14.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 21.4iT - 169T^{2} \)
17 \( 1 + (20.7 - 12.0i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (10.5 + 6.11i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-20.1 + 34.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 26.0T + 841T^{2} \)
31 \( 1 + (21.8 - 12.6i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (6.48 - 11.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 33.8iT - 1.68e3T^{2} \)
43 \( 1 - 29.9T + 1.84e3T^{2} \)
47 \( 1 + (48.2 + 27.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-4.36 - 7.55i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (43.2 - 24.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-3.40 - 1.96i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-52.9 - 91.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 35.0T + 5.04e3T^{2} \)
73 \( 1 + (-40.3 + 23.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (43.4 - 75.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 64.0iT - 6.88e3T^{2} \)
89 \( 1 + (-37.2 - 21.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 28.7iT - 9.40e3T^{2} \)
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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

−12.46220509611873857729569293692, −11.43101294411292797657790031311, −10.57728642023637356910756478460, −9.227257208282072535507598347215, −8.586266361899244082602342828770, −7.20039305207151575044167172711, −6.62093315639874790287386318555, −4.58567073763004957278105495972, −4.06709638020587601354673964256, −2.05237330897291087060328208001, 0.24036364074616699664341298168, 3.09159643606012623314266550482, 3.59546253378493196879985153486, 5.60286401190564712823106429634, 6.47464873229442005176715054355, 7.71275673352822273538202112358, 8.815664537998033280403924197555, 9.494980552904491879546914772891, 11.10823547634148653086064465048, 11.41222220775819080740555577545

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