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Properties

Label 2-1350-9.7-c1-0-3
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

\(L(1)\) \(\approx\) \(1.024279925\)
\(L(\frac12)\) \(\approx\) \(1.024279925\)
\(L(\frac{3}{2})\) 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

−9.748751361065571451630813336316, −9.122313407915733113654369239955, −7.933653287277455323316292036409, −7.63241776608715427184913700853, −6.74042445853328884785243528904, −5.66368460758130718116018695604, −5.05946821199069051270039225008, −3.97720622262018784983392353855, −2.70152647730484150154313557091, −1.28895602039534303826382008075, 0.51270757216369847913234352106, 2.04290339753211947038474179905, 3.00547304902650612275748777160, 4.03436958633878684338116184850, 5.04212092178310158361949310413, 6.01857063879957646581834825497, 6.97537764644931190465912176764, 8.079771546077385340243728800922, 8.467903725557168698115737347824, 9.465371871350710821992060059433

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