Properties

Degree 2
Conductor $ 2^{3} \cdot 7^{2} $
Sign $0.971 - 0.237i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.78 + 0.907i)2-s + (−1.99 − 3.44i)3-s + (2.35 − 3.23i)4-s + (−1.63 − 0.941i)5-s + (6.67 + 4.33i)6-s + (−1.25 + 7.90i)8-s + (−3.42 + 5.93i)9-s + (3.75 + 0.197i)10-s + (3.93 + 6.82i)11-s + (−15.8 − 1.66i)12-s + 11.4i·13-s + 7.49i·15-s + (−4.94 − 15.2i)16-s + (−1.44 − 2.51i)17-s + (0.717 − 13.6i)18-s + (−15.0 + 26.0i)19-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (−0.663 − 1.14i)3-s + (0.587 − 0.808i)4-s + (−0.326 − 0.188i)5-s + (1.11 + 0.722i)6-s + (−0.156 + 0.987i)8-s + (−0.380 + 0.659i)9-s + (0.375 + 0.0197i)10-s + (0.358 + 0.620i)11-s + (−1.31 − 0.138i)12-s + 0.883i·13-s + 0.499i·15-s + (−0.308 − 0.951i)16-s + (−0.0852 − 0.147i)17-s + (0.0398 − 0.759i)18-s + (−0.790 + 1.36i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392\)    =    \(2^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.971 - 0.237i$
motivic weight  =  \(2\)
character  :  $\chi_{392} (275, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 392,\ (\ :1),\ 0.971 - 0.237i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.706437 + 0.0850310i\)
\(L(\frac12)\)  \(\approx\)  \(0.706437 + 0.0850310i\)
\(L(2)\)   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 \( 1 + (1.78 - 0.907i)T \)
7 \( 1 \)
good3 \( 1 + (1.99 + 3.44i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.63 + 0.941i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.93 - 6.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 11.4iT - 169T^{2} \)
17 \( 1 + (1.44 + 2.51i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (15.0 - 26.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-33.3 - 19.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 27.8iT - 841T^{2} \)
31 \( 1 + (-19.4 + 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-39.4 - 22.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 40.6T + 1.68e3T^{2} \)
43 \( 1 + 47.2T + 1.84e3T^{2} \)
47 \( 1 + (-71.5 - 41.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-23.2 + 13.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (5.20 + 9.01i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (19.1 + 11.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-29.6 - 51.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 38.2iT - 5.04e3T^{2} \)
73 \( 1 + (-6.98 - 12.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (44.3 + 25.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 89.4T + 6.88e3T^{2} \)
89 \( 1 + (-52.6 + 91.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 55.3T + 9.40e3T^{2} \)
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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

−11.34311982425693234694317345381, −10.09522934945858757346919346682, −9.219314148510487949957459613277, −8.100022360597484337103922232503, −7.38499447154326943726039981906, −6.53666621912740856701018799470, −5.86159891051947383290699555075, −4.38953188136392116470217180041, −2.11619680820326200646658251052, −0.981323193860938873469372587055, 0.60392361461948579892764315472, 2.81502567479588521232887815195, 3.91162925506755375419330439847, 5.05907843461595645222869806357, 6.37303264545859415468013681159, 7.43639058913027741074955323261, 8.680680171539336848832078922638, 9.262449143469170455959302963762, 10.42666938207102654991705356799, 10.88891495611602909212204688269

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