Properties

Degree 2
Conductor $ 2^{3} \cdot 7^{2} $
Sign $-0.286 + 0.958i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.97 + 0.323i)2-s + (1.70 − 2.95i)3-s + (3.79 − 1.27i)4-s + (−1.34 + 0.774i)5-s + (−2.41 + 6.38i)6-s + (−7.07 + 3.74i)8-s + (−1.32 − 2.30i)9-s + (2.39 − 1.96i)10-s + (2.24 − 3.88i)11-s + (2.70 − 13.3i)12-s + 1.54i·13-s + 5.29i·15-s + (12.7 − 9.66i)16-s + (11.8 − 20.4i)17-s + (3.36 + 4.11i)18-s + (−12.4 − 21.5i)19-s + ⋯
L(s)  = 1  + (−0.986 + 0.161i)2-s + (0.569 − 0.985i)3-s + (0.947 − 0.318i)4-s + (−0.268 + 0.154i)5-s + (−0.402 + 1.06i)6-s + (−0.883 + 0.467i)8-s + (−0.147 − 0.255i)9-s + (0.239 − 0.196i)10-s + (0.203 − 0.353i)11-s + (0.225 − 1.11i)12-s + 0.119i·13-s + 0.352i·15-s + (0.796 − 0.604i)16-s + (0.695 − 1.20i)17-s + (0.186 + 0.228i)18-s + (−0.654 − 1.13i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392\)    =    \(2^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.286 + 0.958i$
motivic weight  =  \(2\)
character  :  $\chi_{392} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 392,\ (\ :1),\ -0.286 + 0.958i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.662521 - 0.889471i\)
\(L(\frac12)\)  \(\approx\)  \(0.662521 - 0.889471i\)
\(L(2)\)   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 \( 1 + (1.97 - 0.323i)T \)
7 \( 1 \)
good3 \( 1 + (-1.70 + 2.95i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.34 - 0.774i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.24 + 3.88i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 1.54iT - 169T^{2} \)
17 \( 1 + (-11.8 + 20.4i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (12.4 + 21.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-30.5 + 17.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 22.4iT - 841T^{2} \)
31 \( 1 + (40.4 + 23.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (50.7 - 29.2i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 26.9T + 1.68e3T^{2} \)
43 \( 1 + 17.1T + 1.84e3T^{2} \)
47 \( 1 + (-31.2 + 18.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (84.7 + 48.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-30.7 + 53.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-32.6 + 18.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-16.6 + 28.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 + (-34.6 + 60.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-33.5 + 19.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 3.61T + 6.88e3T^{2} \)
89 \( 1 + (-22.0 - 38.1i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 96.1T + 9.40e3T^{2} \)
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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

−10.91110450249774552189886355469, −9.629595551408923608122442292643, −8.857752922443501698914231436222, −8.003938288553907538955542276501, −7.17862354294477861587133177623, −6.63785586731138234082859146871, −5.17018245017311139803484951041, −3.17179893582484471589361714285, −2.09122054490692145395137687905, −0.64136042506000896168684520372, 1.58105303880056688812369512005, 3.26528809515673103005807961746, 4.03418178907144871840195362107, 5.62312801357325811393005443804, 6.93464231234156808592954282017, 7.982296953528682148722962637753, 8.795100783867377791589373163062, 9.459423343375504078315364045526, 10.41532597869834545521051741755, 10.83447007182672324360570191051

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