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Properties

Label 2-1350-45.38-c1-0-4
Degree $2$
Conductor $1350$
Sign $0.773 - 0.634i$
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 + (0.521 + 1.94i)7-s + (0.707 − 0.707i)8-s + (1.70 + 0.984i)11-s + (−1.05 + 3.92i)13-s + (1.00 + 1.74i)14-s + (0.500 − 0.866i)16-s + (2.35 + 2.35i)17-s + 3.70i·19-s + (1.90 + 0.509i)22-s + (−6.05 − 1.62i)23-s + 4.06i·26-s + (1.42 + 1.42i)28-s + (3.74 − 6.49i)29-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.197 + 0.736i)7-s + (0.249 − 0.249i)8-s + (0.514 + 0.296i)11-s + (−0.291 + 1.08i)13-s + (0.269 + 0.466i)14-s + (0.125 − 0.216i)16-s + (0.572 + 0.572i)17-s + 0.850i·19-s + (0.405 + 0.108i)22-s + (−1.26 − 0.338i)23-s + 0.797i·26-s + (0.269 + 0.269i)28-s + (0.696 − 1.20i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.773 - 0.634i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.773 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.564888245\)
\(L(\frac12)\) \(\approx\) \(2.564888245\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.521 - 1.94i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.70 - 0.984i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.05 - 3.92i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.35 - 2.35i)T + 17iT^{2} \)
19 \( 1 - 3.70iT - 19T^{2} \)
23 \( 1 + (6.05 + 1.62i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.74 + 6.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.48 - 6.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.26 - 4.26i)T - 37iT^{2} \)
41 \( 1 + (6.13 - 3.54i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.09 + 2.43i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-7.49 + 2.00i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.03 + 7.03i)T - 53iT^{2} \)
59 \( 1 + (1.34 + 2.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.37 - 7.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.18 - 2.19i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.68iT - 71T^{2} \)
73 \( 1 + (1.14 + 1.14i)T + 73iT^{2} \)
79 \( 1 + (10.0 + 5.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.440 + 1.64i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 + (2.60 + 9.71i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.939393793720506169622266082687, −8.825018624933127823228604760424, −8.154519874831785925663812806169, −7.04794772637466712857900102876, −6.25121230212669237197896289313, −5.54506091245079469297751708997, −4.47359565415999633736415473848, −3.79797072979784821891256588639, −2.49046219443720013843931875669, −1.59089973678665925685731614738, 0.890233960230263520619497056192, 2.51599940023026028414328727763, 3.53243817917796885831019510995, 4.38164425921788494376989271034, 5.33401464219444402234464300970, 6.07901901662330809855229912483, 7.15944279982741489033813644270, 7.63529188108073884549680025048, 8.585619654132763346902144880948, 9.585457051045581583982821326215

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