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Properties

Label 2-1350-9.7-c1-0-15
Degree $2$
Conductor $1350$
Sign $-0.710 + 0.703i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.22 − 3.85i)7-s − 0.999·8-s + (0.724 − 1.25i)11-s + (−1.22 − 2.12i)13-s + (−2.22 − 3.85i)14-s + (−0.5 + 0.866i)16-s + 3.89·17-s − 0.550·19-s + (−0.724 − 1.25i)22-s + (1.44 + 2.51i)23-s − 2.44·26-s − 4.44·28-s + (−3 + 5.19i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.840 − 1.45i)7-s − 0.353·8-s + (0.218 − 0.378i)11-s + (−0.339 − 0.588i)13-s + (−0.594 − 1.02i)14-s + (−0.125 + 0.216i)16-s + 0.945·17-s − 0.126·19-s + (−0.154 − 0.267i)22-s + (0.302 + 0.523i)23-s − 0.480·26-s − 0.840·28-s + (−0.557 + 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.710 + 0.703i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.952654704\)
\(L(\frac12)\) \(\approx\) \(1.952654704\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-2.22 + 3.85i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.724 + 1.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 + 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.89T + 17T^{2} \)
19 \( 1 + 0.550T + 19T^{2} \)
23 \( 1 + (-1.44 - 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.72 + 6.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.224 - 0.389i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.44T + 53T^{2} \)
59 \( 1 + (5.62 + 9.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.224 + 0.389i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.72 - 8.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 - 4.79T + 73T^{2} \)
79 \( 1 + (-3.67 + 6.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.510608147506296199647801732429, −8.500092480162761068561682224461, −7.59758278263675647242549461115, −7.03167096340973535448665942877, −5.67256165786379348473394700027, −5.01545305954151297483665052821, −3.94921196887986187882549078793, −3.34359548975723224107693649038, −1.82513405865211832978742413065, −0.73755263789829648905802484506, 1.76971305521092401330458702364, 2.83106908464112750659762663849, 4.12624109501364922686091300810, 5.07488743235353835224645224587, 5.62015064915494750455352286617, 6.57507500041322332597402786402, 7.47461236320493645708503085823, 8.278309740128996352959248120819, 8.961050254231601286232368329591, 9.625481462498857613064610297013

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