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Properties

Label 2-1350-9.4-c1-0-0
Degree $2$
Conductor $1350$
Sign $-0.998 - 0.0561i$
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 + (1.68 + 2.92i)7-s − 0.999·8-s + (−2.18 − 3.78i)11-s + (−3.37 + 5.84i)13-s + (−1.68 + 2.92i)14-s + (−0.5 − 0.866i)16-s − 1.62·17-s − 2.37·19-s + (2.18 − 3.78i)22-s + (−0.686 + 1.18i)23-s − 6.74·26-s − 3.37·28-s + (0.686 + 1.18i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.637 + 1.10i)7-s − 0.353·8-s + (−0.659 − 1.14i)11-s + (−0.935 + 1.61i)13-s + (−0.450 + 0.780i)14-s + (−0.125 − 0.216i)16-s − 0.394·17-s − 0.544·19-s + (0.466 − 0.807i)22-s + (−0.143 + 0.247i)23-s − 1.32·26-s − 0.637·28-s + (0.127 + 0.220i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.998 - 0.0561i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.998 - 0.0561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.155449398\)
\(L(\frac12)\) \(\approx\) \(1.155449398\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 2.92i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.37 - 5.84i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.62T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
23 \( 1 + (0.686 - 1.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.686 - 1.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.37 - 4.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.81 + 4.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.68 - 6.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (-2.18 + 3.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.05 - 7.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 3.11T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.68 + 6.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 + (4.18 + 7.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.786119410953303222133753381929, −8.866731704692558606784359831743, −8.505697753750577879836377048411, −7.53836564486828907623381113602, −6.64360564819188993231464793411, −5.82101939346536012136074021965, −5.05287122772067412600783394698, −4.28606288454581634274843902220, −2.94958009748604298446120992469, −1.96516205169217768083173587473, 0.39325568901143939673284207071, 1.91976876368333965946803393861, 2.89505671910574810782854984780, 4.16250271313971930805068668779, 4.77247838521949474450934730247, 5.58035663198225969691658068112, 6.85939501111415035919907215913, 7.67703162144427077941433231123, 8.208077192333327158054421164463, 9.594793198102994899655294353226

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