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Properties

Label 2-1350-9.7-c1-0-1
Degree $2$
Conductor $1350$
Sign $-0.426 - 0.904i$
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 + (−0.224 + 0.389i)7-s − 0.999·8-s + (−1.72 + 2.98i)11-s + (1.22 + 2.12i)13-s + (0.224 + 0.389i)14-s + (−0.5 + 0.866i)16-s − 5.89·17-s − 5.44·19-s + (1.72 + 2.98i)22-s + (−3.44 − 5.97i)23-s + 2.44·26-s + 0.449·28-s + (−3 + 5.19i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.0849 + 0.147i)7-s − 0.353·8-s + (−0.520 + 0.900i)11-s + (0.339 + 0.588i)13-s + (0.0600 + 0.104i)14-s + (−0.125 + 0.216i)16-s − 1.43·17-s − 1.25·19-s + (0.367 + 0.636i)22-s + (−0.719 − 1.24i)23-s + 0.480·26-s + 0.0849·28-s + (−0.557 + 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.426 - 0.904i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3655225886\)
\(L(\frac12)\) \(\approx\) \(0.3655225886\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.224 - 0.389i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.72 - 2.98i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
19 \( 1 + 5.44T + 19T^{2} \)
23 \( 1 + (3.44 + 5.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.775 + 1.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.27 + 2.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.22 + 3.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.55T + 53T^{2} \)
59 \( 1 + (-6.62 - 11.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.22 - 3.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.27 - 3.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.44T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + (3.67 - 6.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.16474809835555596649182887618, −8.864628865283770518726971330660, −8.703463395664116003135021954056, −7.25144390975825757512644425355, −6.57177226286801055439911592853, −5.60267567417512015858758054547, −4.50725367015567432303227944304, −4.03421978485055092685184925300, −2.54906335586399980911431521724, −1.87622146881427255827401923145, 0.12111231279244567161002258194, 2.13883850801006158997067615313, 3.40519117633578954543157586756, 4.20470070690715993416516927814, 5.29845483134764434110959726166, 6.02919099939187648749648506160, 6.76660385511213378166333206803, 7.76527229948780131405513139685, 8.423482117458747487535307015946, 9.099068681752051470114134395531

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