Properties

Degree $2$
Conductor $2268$
Sign $-0.466 - 0.884i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 1.5i)5-s + (−1.62 − 2.09i)7-s + (−3.67 − 2.12i)11-s + (−2.12 + 1.22i)13-s + 1.73·17-s + 2.44i·19-s + (7.34 − 4.24i)23-s + (1 − 1.73i)25-s + (−3.67 − 2.12i)29-s + (−6.36 + 3.67i)31-s + (−1.73 + 4.24i)35-s + 37-s + (0.866 + 1.5i)41-s + (−3.5 + 6.06i)43-s + (−6.06 + 10.5i)47-s + ⋯
L(s)  = 1  + (−0.387 − 0.670i)5-s + (−0.612 − 0.790i)7-s + (−1.10 − 0.639i)11-s + (−0.588 + 0.339i)13-s + 0.420·17-s + 0.561i·19-s + (1.53 − 0.884i)23-s + (0.200 − 0.346i)25-s + (−0.682 − 0.393i)29-s + (−1.14 + 0.659i)31-s + (−0.292 + 0.717i)35-s + 0.164·37-s + (0.135 + 0.234i)41-s + (−0.533 + 0.924i)43-s + (−0.884 + 1.53i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.466 - 0.884i$
Motivic weight: \(1\)
Character: $\chi_{2268} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.466 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1145606863\)
\(L(\frac12)\) \(\approx\) \(0.1145606863\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (1.62 + 2.09i)T \)
good5 \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.67 + 2.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + (-7.34 + 4.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.67 + 2.12i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.36 - 3.67i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (-0.866 - 1.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.06 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (4.33 + 7.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.12 + 1.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.06 + 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (-2.12 - 1.22i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.296203843902151747038740041439, −8.411909278352080353357214070238, −7.78322227558270424241559002219, −7.05645631240940314010518238337, −6.17770731518342299559540600565, −5.17192132351383543753930766009, −4.54840511109629940628004794667, −3.51698684184372321639618623884, −2.70044341204323468733776347536, −1.11892200331954346720229456668, 0.04265534032201478739561332983, 2.04167612002161852574529267622, 2.96741226093146595557565373121, 3.56459271092644804341032698106, 5.11761990057663727746960960939, 5.34921432415727508516457812874, 6.58044271412466664539854730691, 7.34215314832222424388312145850, 7.72744854730562883607327372597, 8.958117450479426510807583211940

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