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Properties

Label 2-1350-45.4-c1-0-11
Degree $2$
Conductor $1350$
Sign $0.687 + 0.726i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.73 + i)7-s − 0.999i·8-s + (3.46 + 2i)13-s + (−0.999 + 1.73i)14-s + (−0.5 − 0.866i)16-s − 6i·17-s + 7·19-s + 3.99·26-s + 1.99i·28-s + (3 + 5.19i)29-s + (5 − 8.66i)31-s + (−0.866 − 0.499i)32-s + (−3 − 5.19i)34-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.654 + 0.377i)7-s − 0.353i·8-s + (0.960 + 0.554i)13-s + (−0.267 + 0.462i)14-s + (−0.125 − 0.216i)16-s − 1.45i·17-s + 1.60·19-s + 0.784·26-s + 0.377i·28-s + (0.557 + 0.964i)29-s + (0.898 − 1.55i)31-s + (−0.153 − 0.0883i)32-s + (−0.514 − 0.891i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.687 + 0.726i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.687 + 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.375756961\)
\(L(\frac12)\) \(\approx\) \(2.375756961\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.46 - 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.2 + 6.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.79 - 4.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (14.7 - 8.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.384600559969312409072901809031, −9.078639414934169132644247930193, −7.72343295008553425075211558402, −6.93151338559644522924258784023, −6.04350177303251539919452155249, −5.32486250723577321353382844823, −4.30450450655072953135092781700, −3.30045056090137089415652072363, −2.51776144510974271377720287776, −0.981968992972367962785342229785, 1.23058615344226498764497249373, 2.95314355167899729402243025374, 3.62067360650152616557976646609, 4.59748031213442751736779737114, 5.70941964054276276715791347915, 6.28564344775978034721416437487, 7.13119000039801879029430965183, 8.069014632590751208160826642105, 8.669410761440674063936295466714, 9.887084878846757428964970706644

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