Properties

Degree 2
Conductor $ 2 \cdot 19^{2} $
Sign $-0.910 - 0.412i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 7-s − 0.999·8-s + (1 + 1.73i)9-s − 6·11-s + 0.999·12-s + (−2.5 − 4.33i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−1.5 + 2.59i)17-s + 2·18-s + (0.5 − 0.866i)21-s + (−3 + 5.19i)22-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s − 0.377·7-s − 0.353·8-s + (0.333 + 0.577i)9-s − 1.80·11-s + 0.288·12-s + (−0.693 − 1.20i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.363 + 0.630i)17-s + 0.471·18-s + (0.109 − 0.188i)21-s + (−0.639 + 1.10i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(722\)    =    \(2 \cdot 19^{2}\)
\( \varepsilon \)  =  $-0.910 - 0.412i$
motivic weight  =  \(1\)
character  :  $\chi_{722} (429, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  1
Selberg data  =  $(2,\ 722,\ (\ :1/2),\ -0.910 - 0.412i)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;19\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.25765001529881292714042737125, −9.446100495936444896176596385838, −8.092572216291284359554443608429, −7.49553318287091591647380974186, −5.94820683754930139030370328507, −5.23918886731926048487804639182, −4.46515944968277356981876872342, −3.17373830502963259637811322791, −2.20643893612872520195976331227, 0, 2.19454637273018313789251851896, 3.51265975921653664699823417975, 4.79104111464750286928878710427, 5.53479944941460380950923957990, 6.72641471866208949390230551007, 7.12814032116340885760729957749, 8.056905948556871485554174373473, 9.169466048648994520446296461214, 9.886190533135730460502822307684

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