Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.978 + 0.205i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (4.5 − 2.59i)5-s + (−1 + 6.92i)7-s + (4 + 6.92i)9-s + (8.5 − 14.7i)11-s + 13.8i·13-s − 5.19i·15-s + (12.5 − 21.6i)17-s + (−3.5 − 6.06i)19-s + (5.49 + 4.33i)21-s + (4.5 − 2.59i)23-s + (1 − 1.73i)25-s + 17·27-s − 13.8i·29-s + (28.5 + 16.4i)31-s + ⋯
L(s)  = 1  + (0.166 − 0.288i)3-s + (0.900 − 0.519i)5-s + (−0.142 + 0.989i)7-s + (0.444 + 0.769i)9-s + (0.772 − 1.33i)11-s + 1.06i·13-s − 0.346i·15-s + (0.735 − 1.27i)17-s + (−0.184 − 0.319i)19-s + (0.261 + 0.206i)21-s + (0.195 − 0.112i)23-s + (0.0400 − 0.0692i)25-s + 0.629·27-s − 0.477i·29-s + (0.919 + 0.530i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.978 + 0.205i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (207, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.978 + 0.205i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.97797 - 0.205177i\)
\(L(\frac12)\)  \(\approx\)  \(1.97797 - 0.205177i\)
\(L(2)\)   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 \( 1 + (1 - 6.92i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.5 + 2.59i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.5 + 14.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 13.8iT - 169T^{2} \)
17 \( 1 + (-12.5 + 21.6i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-4.5 + 2.59i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 13.8iT - 841T^{2} \)
31 \( 1 + (-28.5 - 16.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (7.5 - 4.33i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 26T + 1.68e3T^{2} \)
43 \( 1 + 14T + 1.84e3T^{2} \)
47 \( 1 + (43.5 - 25.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (79.5 + 45.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (27.5 - 47.6i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (19.5 - 11.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-8.5 + 14.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (59.5 - 103. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-64.5 + 37.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 110T + 6.88e3T^{2} \)
89 \( 1 + (35.5 + 61.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 22T + 9.40e3T^{2} \)
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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

−11.98183841137149583367665519935, −11.22724074280459186907375695752, −9.754337153242918056491548524204, −9.109365318132976808602385263750, −8.229107741081319227184579666243, −6.77296967207349430006778859810, −5.79338405869396939118488398868, −4.75519084767241954599407174703, −2.85692819813363375503043366659, −1.47837986228554648595262899203, 1.49247896338481340627349212567, 3.34837268940314721058804358606, 4.43468771267298504794157420042, 6.06632152671562917702493855584, 6.87289950322865836149543944103, 7.998179545109023527211861883267, 9.565687483787918082469969074042, 10.01878498109662382281425206648, 10.72262484794728323379598385326, 12.29966507003708502875774346575

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