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Properties

Label 2-1350-9.7-c1-0-5
Degree $2$
Conductor $1350$
Sign $0.939 - 0.342i$
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 + 1.73i)7-s − 0.999·8-s + (2 + 3.46i)13-s + (0.999 + 1.73i)14-s + (−0.5 + 0.866i)16-s + 6·17-s − 7·19-s + 3.99·26-s + 1.99·28-s + (−3 + 5.19i)29-s + (5 + 8.66i)31-s + (0.499 + 0.866i)32-s + (3 − 5.19i)34-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.377 + 0.654i)7-s − 0.353·8-s + (0.554 + 0.960i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s + 1.45·17-s − 1.60·19-s + 0.784·26-s + 0.377·28-s + (−0.557 + 0.964i)29-s + (0.898 + 1.55i)31-s + (0.0883 + 0.153i)32-s + (0.514 − 0.891i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.939 - 0.342i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.692424725\)
\(L(\frac12)\) \(\approx\) \(1.692424725\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.742755495446722749277552814935, −8.888413202835885284727811551218, −8.336827757795604578980720960547, −7.02339672376989348738158140264, −6.22266331907481347378149578913, −5.44633561766090119389361912118, −4.41603177722685243977656789804, −3.50795802302985086692142036888, −2.52406634885592162994351918363, −1.35250233327886464128372305174, 0.67414909397168728144384468845, 2.54506878783154788666738785556, 3.73451803868054421804457601815, 4.31827028703831969681873422118, 5.69795143990435600378868565076, 6.05001711473687199327792349644, 7.16646019002409905651346077200, 7.85473308954008222507486573726, 8.503551749929879889468242114389, 9.591481653183311736156714137823

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