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Properties

Label 2-1458-9.4-c1-0-12
Degree $2$
Conductor $1458$
Sign $-1$
Analytic cond. $11.6421$
Root an. cond. $3.41206$
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 + (−1.30 + 2.26i)5-s + (2.56 + 4.44i)7-s − 0.999·8-s − 2.61·10-s + (1.23 + 2.14i)11-s + (−1.90 + 3.29i)13-s + (−2.56 + 4.44i)14-s + (−0.5 − 0.866i)16-s − 2.41·17-s + 5.05·19-s + (−1.30 − 2.26i)20-s + (−1.23 + 2.14i)22-s + (2.85 − 4.95i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.583 + 1.01i)5-s + (0.970 + 1.68i)7-s − 0.353·8-s − 0.825·10-s + (0.373 + 0.646i)11-s + (−0.527 + 0.912i)13-s + (−0.686 + 1.18i)14-s + (−0.125 − 0.216i)16-s − 0.584·17-s + 1.16·19-s + (−0.291 − 0.505i)20-s + (−0.264 + 0.457i)22-s + (0.596 − 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1458\)    =    \(2 \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(11.6421\)
Root analytic conductor: \(3.41206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1458} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1458,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.806457577\)
\(L(\frac12)\) \(\approx\) \(1.806457577\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.30 - 2.26i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.56 - 4.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.23 - 2.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.90 - 3.29i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.41T + 17T^{2} \)
19 \( 1 - 5.05T + 19T^{2} \)
23 \( 1 + (-2.85 + 4.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.81 + 3.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.01 + 3.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.49T + 37T^{2} \)
41 \( 1 + (-5.17 + 8.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.128 - 0.222i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.62 - 6.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 14.4T + 53T^{2} \)
59 \( 1 + (-1.71 + 2.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.80 - 8.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.63 + 2.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.55T + 71T^{2} \)
73 \( 1 - 3.93T + 73T^{2} \)
79 \( 1 + (0.482 + 0.835i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.20 - 3.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.51T + 89T^{2} \)
97 \( 1 + (4.64 + 8.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.592582233509927117100055595912, −9.062206641508077856714791306357, −8.166607724981810818455275804746, −7.39784480051893612237332491953, −6.73529282175927472585852787713, −5.84618300431244096455341061922, −4.92281032236094774450245262663, −4.20303589952357377276887646243, −2.87010421136746023360656311689, −2.09489196785424228486825309386, 0.70772055085242044532541842345, 1.38736755368898708499170327175, 3.20632474210271288672407056420, 3.97270359970701645816316812302, 4.86059567784022487279040782485, 5.28920819111044389483311918899, 6.78770555853325125318500665968, 7.70822580760052398248525482864, 8.196448047083741552505192396881, 9.205942985861235595958990207025

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