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Properties

Label 2-1350-45.4-c1-0-5
Degree $2$
Conductor $1350$
Sign $0.803 - 0.595i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (3.46 − 2i)7-s + 0.999i·8-s + (1.5 + 2.59i)11-s + (3.46 + 2i)13-s + (−1.99 + 3.46i)14-s + (−0.5 − 0.866i)16-s + 3i·17-s − 5·19-s + (−2.59 − 1.5i)22-s + (5.19 + 3i)23-s − 3.99·26-s − 3.99i·28-s + (−3 − 5.19i)29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.30 − 0.755i)7-s + 0.353i·8-s + (0.452 + 0.783i)11-s + (0.960 + 0.554i)13-s + (−0.534 + 0.925i)14-s + (−0.125 − 0.216i)16-s + 0.727i·17-s − 1.14·19-s + (−0.553 − 0.319i)22-s + (1.08 + 0.625i)23-s − 0.784·26-s − 0.755i·28-s + (−0.557 − 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.803 - 0.595i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.523901138\)
\(L(\frac12)\) \(\approx\) \(1.523901138\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-3.46 + 2i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.46 - 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.52 + 5.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 7iT - 73T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 - 6i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (9.52 - 5.5i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.603492053439268916546675490042, −8.710262914531595424410719321908, −8.171534738736043982708540714562, −7.28081967989516007049093331608, −6.64913065466482772560655653547, −5.62608843771465174208185590855, −4.53058597360788282650396521429, −3.88997271371033481726924566915, −2.03344979078046622805162146829, −1.21953593906862902178320874172, 0.948207750971192967279615112620, 2.11322264412609325258323138201, 3.18450156969230141513197225560, 4.34628904067068234527759147299, 5.38711504295753093086838198401, 6.20781854416986378866179027257, 7.29635651898047633334763236148, 8.193742698968795059336039659404, 8.784098645218889060081957167388, 9.196650415688786605195564990920

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