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  :  $\chi_{224} (33, \cdot )$
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

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

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