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Properties

Label 2-1350-45.38-c1-0-12
Degree $2$
Conductor $1350$
Sign $-0.891 + 0.453i$
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 + (−1.18 − 4.40i)7-s + (−0.707 + 0.707i)8-s + (0.550 + 0.317i)11-s + (−0.896 + 3.34i)13-s + (2.28 + 3.94i)14-s + (0.500 − 0.866i)16-s + (−0.317 − 0.317i)17-s − 6.44i·19-s + (−0.614 − 0.164i)22-s + (−0.965 − 0.258i)23-s − 3.46i·26-s + (−3.22 − 3.22i)28-s + (−0.158 + 0.275i)29-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.446 − 1.66i)7-s + (−0.249 + 0.249i)8-s + (0.165 + 0.0958i)11-s + (−0.248 + 0.928i)13-s + (0.609 + 1.05i)14-s + (0.125 − 0.216i)16-s + (−0.0770 − 0.0770i)17-s − 1.47i·19-s + (−0.130 − 0.0350i)22-s + (−0.201 − 0.0539i)23-s − 0.679i·26-s + (−0.609 − 0.609i)28-s + (−0.0295 + 0.0511i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.891 + 0.453i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4524572470\)
\(L(\frac12)\) \(\approx\) \(0.4524572470\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 + 4.40i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.550 - 0.317i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.896 - 3.34i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.317 + 0.317i)T + 17iT^{2} \)
19 \( 1 + 6.44iT - 19T^{2} \)
23 \( 1 + (0.965 + 0.258i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.158 - 0.275i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.224 + 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (6.39 - 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.34 - 0.896i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (8.69 - 2.32i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.78 - 3.78i)T - 53iT^{2} \)
59 \( 1 + (4.48 + 7.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.275 + 0.476i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.38 + 1.71i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.29iT - 71T^{2} \)
73 \( 1 + (-6.89 - 6.89i)T + 73iT^{2} \)
79 \( 1 + (-2.12 - 1.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.41 - 5.26i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 8.02T + 89T^{2} \)
97 \( 1 + (0.695 + 2.59i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.554717091937640389661773241370, −8.440137916827680774858208825090, −7.54521896871139517596395933076, −6.85747553718162507250832337187, −6.43121165451682525443404409805, −4.91526966476426522178072079432, −4.14060108217982491107994445969, −2.99832695652258615603792381640, −1.55173785352322367064097787109, −0.22885987591696831913152970339, 1.69166916809337369770611290610, 2.74688206406923532411762070741, 3.58510920745297359746803374633, 5.15693148555477959049566105679, 5.89557885020375054497253525403, 6.60866074427499679075187006863, 7.86744942297988594500200224438, 8.368683838445090238606456843719, 9.155702933334809938141248106004, 9.886494819604803776112065054561

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