Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 67 $
Sign $-0.978 + 0.205i$
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 + 3-s + (−0.499 − 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + (−1 − 1.73i)7-s − 0.999·8-s − 2·9-s + (−0.5 + 0.866i)10-s + (−3 − 5.19i)11-s + (−0.499 − 0.866i)12-s + (−1 + 1.73i)13-s − 1.99·14-s − 15-s + (−0.5 + 0.866i)16-s + (−3 + 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + 0.577·3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.204 − 0.353i)6-s + (−0.377 − 0.654i)7-s − 0.353·8-s − 0.666·9-s + (−0.158 + 0.273i)10-s + (−0.904 − 1.56i)11-s + (−0.144 − 0.249i)12-s + (−0.277 + 0.480i)13-s − 0.534·14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.727 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(670\)    =    \(2 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.978 + 0.205i$
motivic weight  =  \(1\)
character  :  $\chi_{670} (431, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 670,\ (\ :1/2),\ -0.978 + 0.205i)$
$L(1)$  $\approx$  $0.110885 - 1.06793i$
$L(\frac12)$  $\approx$  $0.110885 - 1.06793i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;67\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
67 \( 1 + (-2.5 - 7.79i)T \)
good3 \( 1 - T + 3T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-7.5 - 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.44842362966500720226932090160, −9.032645143961108496014525540660, −8.635654732808729228666065182018, −7.60937830845796345154928176191, −6.45405413344596892238755058393, −5.44825903193512128852002821382, −4.21766230232826466539162546634, −3.31399820513476425328931648729, −2.48155940693625889045567456367, −0.44634355792084626236331203684, 2.46626714514537242760146275771, 3.23261278261940623949100268241, 4.68449811645055221549683136102, 5.36436203103784840998680545498, 6.54296788073615386345437574074, 7.66829807172261313648704933573, 7.937866989816557168551649806659, 9.272174820517103042001349727963, 9.618538959847183781721636774267, 10.98541347834296746446520843679

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