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Properties

Label 2-1350-45.2-c1-0-15
Degree $2$
Conductor $1350$
Sign $-0.949 + 0.313i$
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 + (−1.67 + 0.448i)7-s + (0.707 + 0.707i)8-s + (−3 − 1.73i)11-s + (3.34 + 0.896i)13-s + (0.866 + 1.50i)14-s + (0.500 − 0.866i)16-s + (4.24 − 4.24i)17-s + 2i·19-s + (−0.896 + 3.34i)22-s + (0.776 − 2.89i)23-s − 3.46i·26-s + (1.22 − 1.22i)28-s + (−4.33 + 7.5i)29-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.632 + 0.169i)7-s + (0.249 + 0.249i)8-s + (−0.904 − 0.522i)11-s + (0.928 + 0.248i)13-s + (0.231 + 0.400i)14-s + (0.125 − 0.216i)16-s + (1.02 − 1.02i)17-s + 0.458i·19-s + (−0.191 + 0.713i)22-s + (0.161 − 0.604i)23-s − 0.679i·26-s + (0.231 − 0.231i)28-s + (−0.804 + 1.39i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.949 + 0.313i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.949 + 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7008976449\)
\(L(\frac12)\) \(\approx\) \(0.7008976449\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 0.448i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.34 - 0.896i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-0.776 + 2.89i)T + (-19.9 - 11.5i)T^{2} \)
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 \( 1 + (2.68 + 10.0i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.776 + 2.89i)T + (-40.7 + 23.5i)T^{2} \)
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 \( 1 + (-3.13 + 11.7i)T + (-58.0 - 33.5i)T^{2} \)
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 \( 1 + (-2.89 + 0.776i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 + (-6.69 + 1.79i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.364224190955663231693177688483, −8.572665863469893811496408234277, −7.79804081347034530599636690277, −6.84657983329672688616057260848, −5.74700769571322441901606670160, −5.07477074897478571979499783008, −3.65077989707617198847631747076, −3.13670357996030676827160867778, −1.85681723996106518761736841303, −0.31281253003120742514611657326, 1.45949974125347788785392875277, 3.09375696866105144897913641469, 3.97634444321583862926620474088, 5.20067791282466590646150652910, 5.85146270500783727973886933643, 6.74224433222839344272298485899, 7.55851727794842256005262882213, 8.249773853234387611241691569124, 9.054593869702212267488367732091, 10.03636115934212700779210138433

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