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Properties

Label 2-1350-45.2-c1-0-2
Degree $2$
Conductor $1350$
Sign $0.695 - 0.718i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−1.22 + 0.328i)7-s + (0.707 + 0.707i)8-s + (−3 − 1.73i)11-s + (1.22 + 0.328i)13-s + (0.633 + 1.09i)14-s + (0.500 − 0.866i)16-s + 7.19i·19-s + (−0.896 + 3.34i)22-s + (2.12 − 7.91i)23-s − 1.26i·26-s + (0.896 − 0.896i)28-s + (−3.63 + 6.29i)29-s + (5.09 + 8.83i)31-s + (−0.965 − 0.258i)32-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.462 + 0.124i)7-s + (0.249 + 0.249i)8-s + (−0.904 − 0.522i)11-s + (0.339 + 0.0910i)13-s + (0.169 + 0.293i)14-s + (0.125 − 0.216i)16-s + 1.65i·19-s + (−0.191 + 0.713i)22-s + (0.442 − 1.65i)23-s − 0.248i·26-s + (0.169 − 0.169i)28-s + (−0.674 + 1.16i)29-s + (0.915 + 1.58i)31-s + (−0.170 − 0.0457i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8882050397\)
\(L(\frac12)\) \(\approx\) \(0.8882050397\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.22 - 0.328i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.22 - 0.328i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 7.19iT - 19T^{2} \)
23 \( 1 + (-2.12 + 7.91i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.63 - 6.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.09 - 8.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.55 + 1.55i)T + 37iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.67 - 6.24i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.55 - 5.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.55 + 1.55i)T + 53iT^{2} \)
59 \( 1 + (-6.23 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.22 - 12.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (-3.67 + 3.67i)T - 73iT^{2} \)
79 \( 1 + (8.66 + 5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.57 - 1.76i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.66T + 89T^{2} \)
97 \( 1 + (-14.1 + 3.79i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.942956481804565461267698383367, −8.794618672992609221014164130673, −8.407495037232314453476095426476, −7.40519321971099637678536054110, −6.34853207870732025264483902397, −5.50379151193817233425352712590, −4.48290893109602690598641107161, −3.39000569025464039184903005348, −2.65007109879057496307145980637, −1.24462637629503166431161775829, 0.42167640344193502250120984428, 2.22624962654204998146827548757, 3.45629927983097129734348234830, 4.58268611119899637069728901921, 5.40615956361264369072914018596, 6.25373606014341619602378586957, 7.19107294201770903393660916382, 7.71098872620722934977260807769, 8.648002849130046361589278615566, 9.563189144398126504106032781604

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