Properties

Degree 2
Conductor $ 2 \cdot 7^{2} $
Sign $-0.266 - 0.963i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(98\)    =    \(2 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.266 - 0.963i$
motivic weight  =  \(1\)
character  :  $\chi_{98} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 98,\ (\ :1/2),\ -0.266 - 0.963i)$
$L(1)$  $\approx$  $0.606661 + 0.797446i$
$L(\frac12)$  $\approx$  $0.606661 + 0.797446i$
$L(\frac{3}{2})$   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

−14.35056284651231564581666825333, −13.39276982253282504579114951402, −12.06926607584256368231314481450, −11.03156301358370933522671055749, −9.969986304007682103457348749805, −8.801779702387547062476919061217, −7.38187572562548851992014849471, −5.90847417571958222976833557124, −4.92836069455357757789022627749, −3.60730141195257487979543238621, 1.47640700840045322049156739130, 3.65733990102187077391429675053, 5.54613123540807352222510231172, 6.53895455765916395067666626447, 7.937843169684972630689982770456, 9.390512869525057287713939002024, 10.83699373690739446285393590903, 11.55908781045891273671686991799, 12.75709997600820319402207612824, 13.12545459994660673294455378565

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