Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.0104 - 0.999i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.19 − 1.84i)3-s + (−2.63 + 1.52i)5-s + (0.812 − 6.95i)7-s + (2.28 + 3.96i)9-s + (−1.17 + 2.03i)11-s + 25.3i·13-s + 11.2·15-s + (3.08 + 1.78i)17-s + (−14.1 + 8.18i)19-s + (−15.4 + 20.6i)21-s + (8.83 + 15.3i)23-s + (−7.85 + 13.6i)25-s + 16.2i·27-s + 36.1·29-s + (6.25 + 3.61i)31-s + ⋯
L(s)  = 1  + (−1.06 − 0.614i)3-s + (−0.527 + 0.304i)5-s + (0.116 − 0.993i)7-s + (0.254 + 0.440i)9-s + (−0.106 + 0.185i)11-s + 1.94i·13-s + 0.748·15-s + (0.181 + 0.104i)17-s + (−0.746 + 0.430i)19-s + (−0.733 + 0.985i)21-s + (0.384 + 0.665i)23-s + (−0.314 + 0.544i)25-s + 0.603i·27-s + 1.24·29-s + (0.201 + 0.116i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0104 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0104 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.0104 - 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.0104 - 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.336038 + 0.339554i\)
\(L(\frac12)\)  \(\approx\)  \(0.336038 + 0.339554i\)
\(L(2)\)   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 \( 1 + (-0.812 + 6.95i)T \)
good3 \( 1 + (3.19 + 1.84i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.63 - 1.52i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (1.17 - 2.03i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 25.3iT - 169T^{2} \)
17 \( 1 + (-3.08 - 1.78i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (14.1 - 8.18i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.83 - 15.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 36.1T + 841T^{2} \)
31 \( 1 + (-6.25 - 3.61i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (18.4 + 31.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 53.7iT - 1.68e3T^{2} \)
43 \( 1 + 51.2T + 1.84e3T^{2} \)
47 \( 1 + (27.1 - 15.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-35.1 + 60.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (81.4 + 47.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.89 - 1.09i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (12.4 - 21.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 50.8T + 5.04e3T^{2} \)
73 \( 1 + (68.9 + 39.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-57.5 - 99.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 154. iT - 6.88e3T^{2} \)
89 \( 1 + (-98.7 + 57.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 53.9iT - 9.40e3T^{2} \)
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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

−12.00184917358118604327992773734, −11.44179953394483830388046488857, −10.67032892842205184436111422751, −9.474379024439299016229065387927, −8.027733160655010902887294153130, −6.92559421550974315638014559623, −6.48836423517176095821281987001, −4.90843007397822242009062665081, −3.77105723851729657613871534109, −1.49587634292256155199659513530, 0.30971118560324316011289760649, 2.90514753192835319204292775890, 4.61544038317919483838371772426, 5.39837305466325051000887032433, 6.32119345755604373255331270449, 8.000715894118996116154302526420, 8.717582815502528415045407283130, 10.22687581926391332976604767969, 10.71997597172306907938659280356, 11.90917976457475221580369839356

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