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Properties

Label 2-1350-9.7-c1-0-14
Degree $2$
Conductor $1350$
Sign $-0.569 + 0.821i$
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 + (2.44 − 4.24i)11-s + (2.22 + 3.85i)13-s + (−2.22 − 3.85i)14-s + (−0.5 + 0.866i)16-s − 4.89·17-s + 2.55·19-s + (−2.44 − 4.24i)22-s + (−1.22 − 2.12i)23-s + 4.44·26-s − 4.44·28-s + (1.22 − 2.12i)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 + (0.738 − 1.27i)11-s + (0.617 + 1.06i)13-s + (−0.594 − 1.02i)14-s + (−0.125 + 0.216i)16-s − 1.18·17-s + 0.585·19-s + (−0.522 − 0.904i)22-s + (−0.255 − 0.442i)23-s + 0.872·26-s − 0.840·28-s + (0.227 − 0.393i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.569 + 0.821i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.098503314\)
\(L(\frac12)\) \(\approx\) \(2.098503314\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 + 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.22 - 3.85i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + (1.22 + 2.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.22 + 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.224 - 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.34T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.72 + 6.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.550 + 0.953i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.44T + 53T^{2} \)
59 \( 1 + (0.275 + 0.476i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.17 + 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.34T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-6.34 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.275 + 0.476i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (5.39 - 9.35i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.302469536433017356923270782851, −8.678905840876421520499159269710, −7.75261432537553152689403109484, −6.71981517267625658596506015499, −6.07598111777561106519912966639, −4.70645263611890698509203204958, −4.16560234923717433500765229917, −3.34707267077599604037004505305, −1.82442962433535292796992759998, −0.817907983125819327648891589277, 1.70906613926827907626348455365, 2.80649797184267824219837101912, 4.11103475144399128030165521335, 4.98700738818076558701185602972, 5.67440497899875455296477728877, 6.50827236564612891639780671963, 7.47095842922707467275342981352, 8.231492749846251399212721963642, 8.983976969166887957325328763604, 9.555922157226026670898614241686

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