Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.899 + 0.437i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.93 − 3.35i)3-s + (−2.33 − 4.05i)5-s + (−6.95 + 0.792i)7-s + (−2.98 − 5.17i)9-s + (−12.6 − 7.29i)11-s + 12.7·13-s − 18.1·15-s + (−16.9 − 9.76i)17-s + (8.86 + 15.3i)19-s + (−10.8 + 24.8i)21-s + (4.43 + 7.67i)23-s + (1.55 − 2.70i)25-s + 11.7·27-s − 35.4i·29-s + (−25.1 − 14.5i)31-s + ⋯
L(s)  = 1  + (0.644 − 1.11i)3-s + (−0.467 − 0.810i)5-s + (−0.993 + 0.113i)7-s + (−0.331 − 0.575i)9-s + (−1.14 − 0.663i)11-s + 0.977·13-s − 1.20·15-s + (−0.994 − 0.574i)17-s + (0.466 + 0.807i)19-s + (−0.514 + 1.18i)21-s + (0.192 + 0.333i)23-s + (0.0623 − 0.108i)25-s + 0.433·27-s − 1.22i·29-s + (−0.810 − 0.468i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.899 + 0.437i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.899 + 0.437i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.267787 - 1.16157i\)
\(L(\frac12)\)  \(\approx\)  \(0.267787 - 1.16157i\)
\(L(2)\)   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 \( 1 + (6.95 - 0.792i)T \)
good3 \( 1 + (-1.93 + 3.35i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (2.33 + 4.05i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (12.6 + 7.29i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 12.7T + 169T^{2} \)
17 \( 1 + (16.9 + 9.76i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-8.86 - 15.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-4.43 - 7.67i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 35.4iT - 841T^{2} \)
31 \( 1 + (25.1 + 14.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-10.5 + 6.10i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 22.0iT - 1.68e3T^{2} \)
43 \( 1 + 79.8iT - 1.84e3T^{2} \)
47 \( 1 + (-36.5 + 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (31.3 + 18.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.20 - 2.08i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (14.6 + 25.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-35.2 - 20.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 22.6T + 5.04e3T^{2} \)
73 \( 1 + (-66.1 - 38.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-68.4 - 118. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 49.9T + 6.88e3T^{2} \)
89 \( 1 + (-0.970 + 0.560i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 158. iT - 9.40e3T^{2} \)
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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

−11.90882195042584324777578525804, −10.73056393911357458751872364214, −9.361096930188530742188216923043, −8.437490552388679893615257195845, −7.78588327266871897216383693395, −6.63874670621114155519808055549, −5.47841872355135910467811001711, −3.75029564174370930537558667557, −2.41330879917902335044275982656, −0.58901567391800494428489719855, 2.82581486268839187780223912171, 3.61335553177342246839346515185, 4.81033148420760223614593715705, 6.43923445372062730597498520301, 7.43942326992054782219443436273, 8.762957663540859513306846104971, 9.518378698938561066293667980401, 10.66366142657854883542496459246, 10.90073432474671507731560895753, 12.61835365493704609919276802072

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