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Properties

Label 2-1470-105.104-c1-0-27
Degree $2$
Conductor $1470$
Sign $0.999 - 0.00978i$
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  − 2-s + (−1.03 − 1.38i)3-s + 4-s + (1.98 + 1.02i)5-s + (1.03 + 1.38i)6-s − 8-s + (−0.844 + 2.87i)9-s + (−1.98 − 1.02i)10-s + 1.27i·11-s + (−1.03 − 1.38i)12-s + 1.32·13-s + (−0.643 − 3.81i)15-s + 16-s − 5.76i·17-s + (0.844 − 2.87i)18-s + 2.44i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.599 − 0.800i)3-s + 0.5·4-s + (0.888 + 0.457i)5-s + (0.423 + 0.566i)6-s − 0.353·8-s + (−0.281 + 0.959i)9-s + (−0.628 − 0.323i)10-s + 0.385i·11-s + (−0.299 − 0.400i)12-s + 0.366·13-s + (−0.166 − 0.986i)15-s + 0.250·16-s − 1.39i·17-s + (0.199 − 0.678i)18-s + 0.560i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.999 - 0.00978i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.999 - 0.00978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.126592677\)
\(L(\frac12)\) \(\approx\) \(1.126592677\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.03 + 1.38i)T \)
5 \( 1 + (-1.98 - 1.02i)T \)
7 \( 1 \)
good11 \( 1 - 1.27iT - 11T^{2} \)
13 \( 1 - 1.32T + 13T^{2} \)
17 \( 1 + 5.76iT - 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + 0.680T + 23T^{2} \)
29 \( 1 + 6.12iT - 29T^{2} \)
31 \( 1 - 6.94iT - 31T^{2} \)
37 \( 1 - 4.67iT - 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 - 1.87iT - 43T^{2} \)
47 \( 1 - 7.97iT - 47T^{2} \)
53 \( 1 + 6.98T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 + 4.22iT - 61T^{2} \)
67 \( 1 - 12.9iT - 67T^{2} \)
71 \( 1 + 5.09iT - 71T^{2} \)
73 \( 1 - 16.4T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + 3.33iT - 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 2.90T + 97T^{2} \)
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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

−9.663101475201918954492413441474, −8.709540733104288476807480529622, −7.73721064030638027964760078007, −7.11795463025417241376149924970, −6.33683130539443849661912147814, −5.72485370406045850584938294199, −4.70849764522369645003502808675, −2.97360295381570568542584461075, −2.10114403533156691246294503647, −1.00988876046762806493764280032, 0.78254567530567599785394652482, 2.11689007524147688179338599379, 3.50071039676040608720315310209, 4.50663239714620709257376223638, 5.65596940212675835309710231254, 6.01396321384290191394839588597, 6.96549532258274026578689631481, 8.242069264232254375749592701298, 8.891877319260579652055100339380, 9.500748295716768128669682824565

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