Properties

Degree 2
Conductor $ 2 \cdot 7^{2} $
Sign $0.991 - 0.126i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + 1.99·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.999 + 1.73i)12-s − 4·13-s + (−0.5 − 0.866i)16-s + (−3 + 5.19i)17-s + (0.499 − 0.866i)18-s + (−1 − 1.73i)19-s + (−0.999 + 1.73i)24-s + (2.5 − 4.33i)25-s + (−2 − 3.46i)26-s + 4.00·27-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + 0.816·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.288 + 0.499i)12-s − 1.10·13-s + (−0.125 − 0.216i)16-s + (−0.727 + 1.26i)17-s + (0.117 − 0.204i)18-s + (−0.229 − 0.397i)19-s + (−0.204 + 0.353i)24-s + (0.5 − 0.866i)25-s + (−0.392 − 0.679i)26-s + 0.769·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(98\)    =    \(2 \cdot 7^{2}\)
\( \varepsilon \)  =  $0.991 - 0.126i$
motivic weight  =  \(1\)
character  :  $\chi_{98} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 98,\ (\ :1/2),\ 0.991 - 0.126i)$
$L(1)$  $\approx$  $1.29448 + 0.0821484i$
$L(\frac12)$  $\approx$  $1.29448 + 0.0821484i$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−14.00599460194856240485032040018, −12.89897144252259646211093501170, −12.44533047287876342504227065817, −10.84462218398902176922880365263, −9.241789904092050155683227766971, −8.083333619699282901638240448876, −7.25391899305612172511439282965, −6.12510237641322476348159870666, −4.42448063842485448054280162476, −2.41749920092248519897743465838, 2.71904573650174489385286764559, 4.12098676425324769033179756617, 5.23488858771214743865463879013, 7.16393847545088947450776580615, 8.904163825273414140536527508881, 9.630199992252261854634945255997, 10.60537085923855477135106016776, 11.74368042716781372037397857686, 12.85516300122271479135410455197, 14.04145501722198037479383964324

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