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Properties

Label 2-1350-45.32-c1-0-8
Degree $2$
Conductor $1350$
Sign $-0.144 - 0.989i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.448 + 1.67i)7-s + (0.707 + 0.707i)8-s + (−3 + 1.73i)11-s + (0.896 + 3.34i)13-s + (−0.866 + 1.50i)14-s + (0.500 + 0.866i)16-s + (4.24 − 4.24i)17-s + 2i·19-s + (−3.34 + 0.896i)22-s + (−2.89 + 0.776i)23-s + 3.46i·26-s + (−1.22 + 1.22i)28-s + (4.33 + 7.5i)29-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (−0.169 + 0.632i)7-s + (0.249 + 0.249i)8-s + (−0.904 + 0.522i)11-s + (0.248 + 0.928i)13-s + (−0.231 + 0.400i)14-s + (0.125 + 0.216i)16-s + (1.02 − 1.02i)17-s + 0.458i·19-s + (−0.713 + 0.191i)22-s + (−0.604 + 0.161i)23-s + 0.679i·26-s + (−0.231 + 0.231i)28-s + (0.804 + 1.39i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.131421198\)
\(L(\frac12)\) \(\approx\) \(2.131421198\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.448 - 1.67i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.896 - 3.34i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (2.89 - 0.776i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.33 - 7.5i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 + (7.5 + 4.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.0 + 2.68i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.89 - 0.776i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (-3.46 + 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.7 + 3.13i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-9.79 + 9.79i)T - 73iT^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.776 - 2.89i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + (-1.79 + 6.69i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.910678753963055529219089062594, −8.940840719753208598587206936408, −8.158189031974170490327698869900, −7.17502848695235216922626179619, −6.59911326892722241395658989195, −5.32535972727953888410935085656, −5.10266833137970634545389301723, −3.74890591802656943055662642400, −2.86987175340067480927822931456, −1.73507812412741950231261253609, 0.68446412509463942769998929382, 2.28514483977314250955482229794, 3.37141523751221387611495232713, 4.07625534207241152146562379361, 5.27567399992381067987337695698, 5.88101088093123211124043004151, 6.77185289601499423641598833934, 8.019389287840183421059756300291, 8.121903135484913507030204749280, 9.771587265585226953232215932090

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