Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.560 + 0.828i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.91 − 0.563i)2-s + (−4.49 − 1.86i)3-s + (3.36 + 2.16i)4-s + (0.432 + 1.04i)5-s + (7.57 + 6.10i)6-s + (−6.93 − 0.951i)7-s + (−5.23 − 6.04i)8-s + (10.3 + 10.3i)9-s + (−0.241 − 2.24i)10-s + (4.35 + 10.5i)11-s + (−11.0 − 15.9i)12-s + (−5.75 + 13.8i)13-s + (12.7 + 5.73i)14-s − 5.49i·15-s + (6.63 + 14.5i)16-s + 2.83·17-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (−1.49 − 0.620i)3-s + (0.841 + 0.540i)4-s + (0.0864 + 0.208i)5-s + (1.26 + 1.01i)6-s + (−0.990 − 0.135i)7-s + (−0.654 − 0.756i)8-s + (1.15 + 1.15i)9-s + (−0.0241 − 0.224i)10-s + (0.396 + 0.956i)11-s + (−0.924 − 1.33i)12-s + (−0.442 + 1.06i)13-s + (0.912 + 0.409i)14-s − 0.366i·15-s + (0.414 + 0.909i)16-s + 0.166·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.560 + 0.828i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.560 + 0.828i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.398576 - 0.211597i\)
\(L(\frac12)\)  \(\approx\)  \(0.398576 - 0.211597i\)
\(L(2)\)   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 \( 1 + (1.91 + 0.563i)T \)
7 \( 1 + (6.93 + 0.951i)T \)
good3 \( 1 + (4.49 + 1.86i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-0.432 - 1.04i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (-4.35 - 10.5i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (5.75 - 13.8i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 2.83T + 289T^{2} \)
19 \( 1 + (-11.1 + 26.9i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (18.7 + 18.7i)T + 529iT^{2} \)
29 \( 1 + (-0.210 + 0.507i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 35.8iT - 961T^{2} \)
37 \( 1 + (-59.9 + 24.8i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (20.9 - 20.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (-23.1 - 55.8i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 2.87T + 2.20e3T^{2} \)
53 \( 1 + (12.1 + 29.2i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-23.7 - 57.4i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-48.9 - 20.2i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-18.8 + 45.5i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-40.7 + 40.7i)T - 5.04e3iT^{2} \)
73 \( 1 + (-59.6 + 59.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 89.6iT - 6.24e3T^{2} \)
83 \( 1 + (23.1 - 55.9i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (32.4 + 32.4i)T + 7.92e3iT^{2} \)
97 \( 1 - 29.3iT - 9.40e3T^{2} \)
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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

−11.76959080703312963343331512312, −11.01908204268664288928899572591, −9.942532952714143536900441448209, −9.311138239700623435192069543299, −7.55947977017298047445033081488, −6.72885194246743635574127833493, −6.24843162450782304339344900595, −4.46482516936937399568601772946, −2.36552077815957497370923457144, −0.61872696449898657054845784796, 0.821299991847207795286261765312, 3.42511719029052523760371635828, 5.47422360897613714384299571103, 5.84232639081122407782899783450, 6.96222964532648972006668329991, 8.310175627383706784749319150917, 9.636211461571967889934790348031, 10.10740534932116809866742424693, 11.01476465254744205089009779765, 11.91147006176413316740260481275

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