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Properties

Label 2-1350-9.4-c1-0-1
Degree $2$
Conductor $1350$
Sign $0.766 - 0.642i$
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 + (−0.5 − 0.866i)7-s + 0.999·8-s + (−1 − 1.73i)11-s + (−3 + 5.19i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 2·17-s + 6·19-s + (−0.999 + 1.73i)22-s + (−0.5 + 0.866i)23-s + 6·26-s + 0.999·28-s + (4.5 + 7.79i)29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (−0.301 − 0.522i)11-s + (−0.832 + 1.44i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.485·17-s + 1.37·19-s + (−0.213 + 0.369i)22-s + (−0.104 + 0.180i)23-s + 1.17·26-s + 0.188·28-s + (0.835 + 1.44i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9400675593\)
\(L(\frac12)\) \(\approx\) \(0.9400675593\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (5.5 - 9.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.5 - 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.705597787955224764781128335724, −9.057008181439197973798252655963, −8.242687736498267614239418014754, −7.24046919904606852576774217420, −6.67642431260010941824577541078, −5.34015872045653045821924885790, −4.49496294223396385656924773829, −3.46086817438515889622005409685, −2.49531639948380161260716019037, −1.21372219481813204100835639292, 0.47950114114859355543179917953, 2.23355680164371202960505269091, 3.31505813740032870256369048162, 4.74325530090970213823171592608, 5.36148785071578602321403762230, 6.24843427056678073872034236478, 7.24294825168603163060247173555, 7.81660977535725662027435530836, 8.600392666216285463013446302870, 9.584827168321639797826599466471

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