Properties

Degree 2
Conductor $ 2^{4} \cdot 43 $
Sign $0.819 + 0.572i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 2.26i)3-s + (−0.618 + 1.07i)5-s + (−2.11 − 3.66i)7-s + (−1.92 − 3.33i)9-s + 3.61·11-s + (−0.690 − 1.19i)13-s + (−1.61 − 2.80i)15-s + (−3.04 − 5.27i)17-s + (−0.618 + 1.07i)19-s + 11.0·21-s + (2.19 − 3.79i)23-s + (1.73 + 3.00i)25-s + 2.23·27-s + (−1.5 − 2.59i)29-s + (−4.73 + 8.20i)33-s + ⋯
L(s)  = 1  + (−0.755 + 1.30i)3-s + (−0.276 + 0.478i)5-s + (−0.800 − 1.38i)7-s + (−0.642 − 1.11i)9-s + 1.09·11-s + (−0.191 − 0.331i)13-s + (−0.417 − 0.723i)15-s + (−0.738 − 1.27i)17-s + (−0.141 + 0.245i)19-s + 2.42·21-s + (0.456 − 0.791i)23-s + (0.347 + 0.601i)25-s + 0.430·27-s + (−0.278 − 0.482i)29-s + (−0.824 + 1.42i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(688\)    =    \(2^{4} \cdot 43\)
\( \varepsilon \)  =  $0.819 + 0.572i$
motivic weight  =  \(1\)
character  :  $\chi_{688} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 688,\ (\ :1/2),\ 0.819 + 0.572i)\)
\(L(1)\)  \(\approx\)  \(0.704562 - 0.221746i\)
\(L(\frac12)\)  \(\approx\)  \(0.704562 - 0.221746i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;43\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 + (-6.5 + 0.866i)T \)
good3 \( 1 + (1.30 - 2.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.618 - 1.07i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.11 + 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.61T + 11T^{2} \)
13 \( 1 + (0.690 + 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.04 + 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.618 - 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.19 + 3.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.42 + 4.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.47T + 41T^{2} \)
47 \( 1 + 1.14T + 47T^{2} \)
53 \( 1 + (-0.690 + 1.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + (1.42 + 2.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.92 - 3.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.39 + 9.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.927 + 1.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.690 + 1.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.01 + 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.927 + 1.60i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.23T + 97T^{2} \)
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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

−10.60277377968420115264712695571, −9.566943693281773361733834232528, −9.189062411617983792329852104560, −7.49937664576005641230588482778, −6.81515685889756900098772470019, −5.91070079527351790317020071581, −4.57575446471385514120899861455, −4.05807445298138916146144196047, −3.07283095897408718613083925455, −0.47272693966423748827090238065, 1.30822638810636183787085016795, 2.52375011504096944974253807479, 4.11622016275687595639130547902, 5.47106484423715376105688397300, 6.28139288400936353671126413303, 6.72049559043565511753604906659, 7.891894758549216443089995967774, 8.890345476596977233684917399964, 9.374809226238178531875003019704, 10.88191088824694637019176931798

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