Properties

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

Origins

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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  :  $\chi_{124} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 124,\ (\ :0),\ 0.390 - 0.920i)$
$L(\frac{1}{2})$  $\approx$  $0.6137468847$
$L(\frac12)$  $\approx$  $0.6137468847$
$L(1)$   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

−13.90200077274478395209904132357, −13.18642218246635535579533856888, −12.17674600461428497912601104499, −10.25696943349034234137116405243, −9.326942655554410349488623302761, −8.536369759493712591970660613850, −7.55840518286483759960699494642, −6.18483581618578829344299054578, −4.55972529911887285331881387887, −3.62548533295285780113509540828, 2.76326840611427196538532991642, 3.23664863338373525323008173673, 5.41607412047591697559067547569, 7.21735457249255791734349745664, 8.302147952705361882193863854075, 9.265677000862623320034465975496, 10.54397566200736196196048454036, 11.32360421178795386961626160752, 12.59387178164138923123330030976, 13.39856948619495352602979483992

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