Properties

Degree 2
Conductor $ 2^{2} \cdot 31 $
Sign $-0.390 - 0.920i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s i·8-s + (−0.866 − 0.5i)10-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.866 − 0.5i)7-s i·8-s + (−0.866 − 0.5i)10-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s − 0.999i·15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(124\)    =    \(2^{2} \cdot 31\)
\( \varepsilon \)  =  $-0.390 - 0.920i$
motivic weight  =  \(0\)
character  :  $\chi_{124} (87, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 124,\ (\ :0),\ -0.390 - 0.920i)$
$L(\frac{1}{2})$  $\approx$  $0.4496221474$
$L(\frac12)$  $\approx$  $0.4496221474$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;31\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - iT \)
31 \( 1 + iT \)
good3 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−14.38064540906110877033278196375, −13.18153866567090037113711101746, −11.67379482032757640573734427078, −10.90414473034916737584421224431, −9.935290020569122213739987438421, −8.387361065421477009127377174460, −7.35518884853854989998619678137, −6.30474776983992920335379169545, −5.01178891049608934019955746078, −3.94693333798037946909518849036, 1.52908883261400833847866302720, 4.01588990366557451481704433847, 5.15770407288810986410585864163, 6.51577319627012882774980448639, 8.622440126044797827118706350253, 8.768357939163126364910945396664, 10.69380714326108650488253231074, 11.61865458042174095040592607421, 11.99553861310626440190086689809, 12.91837495169102241744881554199

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