Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.870 + 0.492i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.824 − 1.42i)3-s + (3.95 − 2.28i)5-s + (6.75 + 1.83i)7-s + (3.14 + 5.43i)9-s + (−6.18 + 10.7i)11-s − 18.3i·13-s − 7.52i·15-s + (6.51 − 11.2i)17-s + (−1.51 − 2.61i)19-s + (8.18 − 8.13i)21-s + (26.2 − 15.1i)23-s + (−2.09 + 3.63i)25-s + 25.1·27-s + 22.7i·29-s + (−19.5 − 11.2i)31-s + ⋯
L(s)  = 1  + (0.274 − 0.475i)3-s + (0.790 − 0.456i)5-s + (0.965 + 0.262i)7-s + (0.348 + 0.604i)9-s + (−0.562 + 0.974i)11-s − 1.41i·13-s − 0.501i·15-s + (0.383 − 0.663i)17-s + (−0.0796 − 0.137i)19-s + (0.389 − 0.387i)21-s + (1.14 − 0.659i)23-s + (−0.0839 + 0.145i)25-s + 0.933·27-s + 0.785i·29-s + (−0.630 − 0.363i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.870 + 0.492i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (207, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.870 + 0.492i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.03072 - 0.535137i\)
\(L(\frac12)\)  \(\approx\)  \(2.03072 - 0.535137i\)
\(L(2)\)   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 \( 1 + (-6.75 - 1.83i)T \)
good3 \( 1 + (-0.824 + 1.42i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.95 + 2.28i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (6.18 - 10.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 18.3iT - 169T^{2} \)
17 \( 1 + (-6.51 + 11.2i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (1.51 + 2.61i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-26.2 + 15.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 22.7iT - 841T^{2} \)
31 \( 1 + (19.5 + 11.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.9 + 6.88i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 60.5T + 1.68e3T^{2} \)
43 \( 1 + 39.0T + 1.84e3T^{2} \)
47 \( 1 + (-17.6 + 10.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-4.12 - 2.38i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-5.86 + 10.1i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (94.3 - 54.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (39.5 - 68.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 12.9iT - 5.04e3T^{2} \)
73 \( 1 + (-49.2 + 85.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (113. - 65.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 28.3T + 6.88e3T^{2} \)
89 \( 1 + (-78.7 - 136. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 39.6T + 9.40e3T^{2} \)
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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

−12.20361361157863225576958343995, −10.83506098626902858408133775141, −10.08059193636000756091900093252, −8.889633825267240025676891962367, −7.908611463690529153015270559521, −7.13843317390929738254608326084, −5.39199311654564423324809616025, −4.89602454468035097586213032394, −2.65867871163712756028765098453, −1.46634765119382521410845708182, 1.67326049554080015315210636667, 3.34922010994147432579079615024, 4.62019789904522893638515368839, 5.91246979722341044817880388599, 6.98964442851767650136113318324, 8.302670054254598760654376659293, 9.234446316767541394567944393988, 10.18465819735422451136648722014, 11.01257182076370299702796680089, 11.92479853463834360626407560872

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