Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.0550 - 0.998i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.33 − 2.5i)3-s + (−4.5 − 7.79i)5-s + (−6.92 − i)7-s + (8.00 + 13.8i)9-s + (−2.59 − 1.5i)11-s + 16·13-s + 45.0i·15-s + (3.5 − 6.06i)17-s + (−9.52 + 5.5i)19-s + (27.5 + 21.6i)21-s + (16.4 − 9.5i)23-s + (−28 + 48.4i)25-s − 35.0i·27-s − 32·29-s + (−9.52 − 5.5i)31-s + ⋯
L(s)  = 1  + (−1.44 − 0.833i)3-s + (−0.900 − 1.55i)5-s + (−0.989 − 0.142i)7-s + (0.888 + 1.53i)9-s + (−0.236 − 0.136i)11-s + 1.23·13-s + 3.00i·15-s + (0.205 − 0.356i)17-s + (−0.501 + 0.289i)19-s + (1.30 + 1.03i)21-s + (0.715 − 0.413i)23-s + (−1.12 + 1.93i)25-s − 1.29i·27-s − 1.10·29-s + (−0.307 − 0.177i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0550 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0550 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.0550 - 0.998i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (95, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.0550 - 0.998i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0617661 + 0.0652652i\)
\(L(\frac12)\)  \(\approx\)  \(0.0617661 + 0.0652652i\)
\(L(2)\)   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 \( 1 + (6.92 + i)T \)
good3 \( 1 + (4.33 + 2.5i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (4.5 + 7.79i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 16T + 169T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (9.52 - 5.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-16.4 + 9.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 32T + 841T^{2} \)
31 \( 1 + (9.52 + 5.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 40T + 1.68e3T^{2} \)
43 \( 1 - 40iT - 1.84e3T^{2} \)
47 \( 1 + (-73.6 + 42.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (3.5 - 6.06i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (45.8 + 26.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (39.5 + 68.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-9.52 - 5.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 48iT - 5.04e3T^{2} \)
73 \( 1 + (71.5 - 123. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-30.3 + 17.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 8iT - 6.88e3T^{2} \)
89 \( 1 + (-48.5 - 84.0i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 88T + 9.40e3T^{2} \)
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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

−11.51934982641402996625723737763, −10.71168384463905913158750938379, −9.238265285031001028746504171332, −8.207815022227149660868329732131, −7.12262225488378397405422458769, −6.03548535236801362009784762212, −5.15518766003113898153590783064, −3.87853562400416745802203871703, −1.15177743522728845530874508837, −0.06974550543499209033988254261, 3.25500969137098953636602295320, 4.09628664832607162624673477700, 5.74260047073204042903818080100, 6.47698602962368446152972701148, 7.38402988580483885822651657793, 9.091364774390166897572627246072, 10.40812505859048670367585911849, 10.72387635628108281440353013918, 11.49873531663241467586324012731, 12.37032624838753762223221091507

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