Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.980 + 0.197i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)5-s − 7i·7-s + (−4 + 6.92i)9-s + (14.7 − 8.5i)11-s + 24·13-s − 0.999i·15-s + (−0.5 − 0.866i)17-s + (−6.06 − 3.5i)19-s + (3.5 + 6.06i)21-s + (6.06 + 3.5i)23-s + (12 + 20.7i)25-s − 17i·27-s + 24·29-s + (35.5 − 20.5i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.166i)3-s + (−0.100 + 0.173i)5-s i·7-s + (−0.444 + 0.769i)9-s + (1.33 − 0.772i)11-s + 1.84·13-s − 0.0666i·15-s + (−0.0294 − 0.0509i)17-s + (−0.319 − 0.184i)19-s + (0.166 + 0.288i)21-s + (0.263 + 0.152i)23-s + (0.479 + 0.831i)25-s − 0.629i·27-s + 0.827·29-s + (1.14 − 0.661i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.980 + 0.197i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (191, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.980 + 0.197i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.49772 - 0.149081i\)
\(L(\frac12)\)  \(\approx\)  \(1.49772 - 0.149081i\)
\(L(2)\)   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 \( 1 + 7iT \)
good3 \( 1 + (0.866 - 0.5i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-14.7 + 8.5i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 24T + 169T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (6.06 + 3.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-6.06 - 3.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 24T + 841T^{2} \)
31 \( 1 + (-35.5 + 20.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-24.5 + 42.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 48T + 1.68e3T^{2} \)
43 \( 1 - 24iT - 1.84e3T^{2} \)
47 \( 1 + (47.6 + 27.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-12.5 - 21.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (14.7 - 8.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (56.2 - 32.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 96iT - 5.04e3T^{2} \)
73 \( 1 + (47.5 + 82.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (35.5 + 20.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 72iT - 6.88e3T^{2} \)
89 \( 1 + (47.5 - 82.2i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 144T + 9.40e3T^{2} \)
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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

−11.55731188619678256431839426369, −11.15639124984579661411175612411, −10.31423194653234954216483809557, −8.943954435369567798588304496331, −8.124221418843820805021395188875, −6.76453389539697833731534766449, −5.92970404734542269380602983807, −4.39927744703282405432001409888, −3.36540246700315755760902864489, −1.12026154097295026105710592308, 1.33638206571630237313296692738, 3.26141490713238760708481507861, 4.62780547793922233015425645479, 6.21939568193788991521427275247, 6.52037659626254115318793603819, 8.480687416624560540349841719584, 8.852471712817776889468075022307, 10.07783422942464561078495221759, 11.44122853950036505022272838182, 11.95931408588974787273697511962

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