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  + (−0.190 + 0.330i)3-s + (1.61 − 2.80i)5-s + (0.118 + 0.204i)7-s + (1.42 + 2.47i)9-s + 1.38·11-s + (−1.80 − 3.13i)13-s + (0.618 + 1.07i)15-s + (2.54 + 4.40i)17-s + (1.61 − 2.80i)19-s − 0.0901·21-s + (3.30 − 5.73i)23-s + (−2.73 − 4.73i)25-s − 2.23·27-s + (−1.5 − 2.59i)29-s + (−0.263 + 0.457i)33-s + ⋯
L(s)  = 1  + (−0.110 + 0.190i)3-s + (0.723 − 1.25i)5-s + (0.0446 + 0.0772i)7-s + (0.475 + 0.823i)9-s + 0.416·11-s + (−0.501 − 0.869i)13-s + (0.159 + 0.276i)15-s + (0.617 + 1.06i)17-s + (0.371 − 0.642i)19-s − 0.0196·21-s + (0.689 − 1.19i)23-s + (−0.547 − 0.947i)25-s − 0.430·27-s + (−0.278 − 0.482i)29-s + (−0.0459 + 0.0795i)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  =  \((2,\ 688,\ (\ :1/2),\ 0.819 + 0.572i)\)
\(L(1)\)  \(\approx\)  \(1.65531 - 0.520974i\)
\(L(\frac12)\)  \(\approx\)  \(1.65531 - 0.520974i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.190 - 0.330i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.61 + 2.80i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.118 - 0.204i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + (1.80 + 3.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.54 - 4.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.61 + 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.30 + 5.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.927 - 1.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.527T + 41T^{2} \)
47 \( 1 + 7.85T + 47T^{2} \)
53 \( 1 + (-1.80 + 3.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.09T + 59T^{2} \)
61 \( 1 + (-1.92 - 3.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.42 + 2.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.89 - 11.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.42 - 4.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.80 + 3.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.51 - 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.42 - 4.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.76T + 97T^{2} \)
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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

−10.21197798556443017749219806996, −9.639809406109778922914826254467, −8.643743749014261947212206554463, −8.004956536925701137209806495492, −6.82682395476057793983019899233, −5.56505670930235103691182477619, −5.08136394797352630081162618916, −4.07103866708086323633214627811, −2.40833198999483307811145547683, −1.10196448655760376159716388555, 1.49855150716976584606688419146, 2.86527865341171090174916327002, 3.84481178045795871696941462992, 5.28389221529602260715502623056, 6.28381117976259863957200415080, 7.01813724930605880428625763430, 7.55878603661399128159210029820, 9.312272202013618493701678330298, 9.539874404944044806954878278269, 10.49513866887602080179352574302

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