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Properties

Label 2-1470-21.20-c1-0-38
Degree $2$
Conductor $1470$
Sign $-0.971 - 0.238i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
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  − i·2-s + (−1.72 − 0.144i)3-s − 4-s − 5-s + (−0.144 + 1.72i)6-s + i·8-s + (2.95 + 0.498i)9-s + i·10-s + 5.90i·11-s + (1.72 + 0.144i)12-s − 1.19i·13-s + (1.72 + 0.144i)15-s + 16-s + 0.519·17-s + (0.498 − 2.95i)18-s − 5.46i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.996 − 0.0833i)3-s − 0.5·4-s − 0.447·5-s + (−0.0589 + 0.704i)6-s + 0.353i·8-s + (0.986 + 0.166i)9-s + 0.316i·10-s + 1.78i·11-s + (0.498 + 0.0416i)12-s − 0.330i·13-s + (0.445 + 0.0372i)15-s + 0.250·16-s + 0.126·17-s + (0.117 − 0.697i)18-s − 1.25i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.971 - 0.238i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.971 - 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2386327593\)
\(L(\frac12)\) \(\approx\) \(0.2386327593\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.72 + 0.144i)T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 - 5.90iT - 11T^{2} \)
13 \( 1 + 1.19iT - 13T^{2} \)
17 \( 1 - 0.519T + 17T^{2} \)
19 \( 1 + 5.46iT - 19T^{2} \)
23 \( 1 + 6.81iT - 23T^{2} \)
29 \( 1 - 6.27iT - 29T^{2} \)
31 \( 1 + 3.20iT - 31T^{2} \)
37 \( 1 + 4.15T + 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 - 5.45T + 47T^{2} \)
53 \( 1 + 4.12iT - 53T^{2} \)
59 \( 1 - 2.16T + 59T^{2} \)
61 \( 1 - 9.24iT - 61T^{2} \)
67 \( 1 + 7.37T + 67T^{2} \)
71 \( 1 + 9.94iT - 71T^{2} \)
73 \( 1 - 9.04iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 2.22T + 89T^{2} \)
97 \( 1 + 9.01iT - 97T^{2} \)
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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

−9.280490598698446153310730529415, −8.368607880319728310951911089854, −7.14936288430065768578730689376, −6.89924900973328190351405203511, −5.51295862810639841490393085043, −4.70753264284989549794150657583, −4.18670051056197073445291751517, −2.75205848176641291342403609867, −1.57099076177057160170458766328, −0.12181769599622481448303920096, 1.29085556857851338640145725127, 3.40129587264358696765519066980, 4.08135915076819320774334898710, 5.29914313650715008360903594616, 5.80779006436508349683316567488, 6.54485655977332899386591881879, 7.47664089662336974064853704631, 8.195414426547530365725427120500, 9.010251844133829260250976922951, 9.990340911080144958708407552443

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