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Properties

Label 2-1470-35.13-c1-0-12
Degree $2$
Conductor $1470$
Sign $0.241 - 0.970i$
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  + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (1.49 + 1.66i)5-s − 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−2.23 − 0.122i)10-s − 0.520·11-s + (0.707 + 0.707i)12-s + (2.39 − 2.39i)13-s + (−2.23 − 0.122i)15-s − 1.00·16-s + (−0.110 − 0.110i)17-s + (0.707 + 0.707i)18-s + 6.73·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.667 + 0.744i)5-s − 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (−0.706 − 0.0386i)10-s − 0.156·11-s + (0.204 + 0.204i)12-s + (0.663 − 0.663i)13-s + (−0.576 − 0.0315i)15-s − 0.250·16-s + (−0.0267 − 0.0267i)17-s + (0.166 + 0.166i)18-s + 1.54·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.241 - 0.970i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.344605879\)
\(L(\frac12)\) \(\approx\) \(1.344605879\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.49 - 1.66i)T \)
7 \( 1 \)
good11 \( 1 + 0.520T + 11T^{2} \)
13 \( 1 + (-2.39 + 2.39i)T - 13iT^{2} \)
17 \( 1 + (0.110 + 0.110i)T + 17iT^{2} \)
19 \( 1 - 6.73T + 19T^{2} \)
23 \( 1 + (-0.802 - 0.802i)T + 23iT^{2} \)
29 \( 1 - 1.20iT - 29T^{2} \)
31 \( 1 + 7.18iT - 31T^{2} \)
37 \( 1 + (-4.41 + 4.41i)T - 37iT^{2} \)
41 \( 1 + 1.23iT - 41T^{2} \)
43 \( 1 + (-6.27 - 6.27i)T + 43iT^{2} \)
47 \( 1 + (-7.57 - 7.57i)T + 47iT^{2} \)
53 \( 1 + (-0.550 - 0.550i)T + 53iT^{2} \)
59 \( 1 + 8.93T + 59T^{2} \)
61 \( 1 + 15.4iT - 61T^{2} \)
67 \( 1 + (10.4 - 10.4i)T - 67iT^{2} \)
71 \( 1 - 3.05T + 71T^{2} \)
73 \( 1 + (-3.22 + 3.22i)T - 73iT^{2} \)
79 \( 1 - 2.29iT - 79T^{2} \)
83 \( 1 + (3.43 - 3.43i)T - 83iT^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 + (-9.40 - 9.40i)T + 97iT^{2} \)
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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

−9.500932756695375498729409193733, −9.220952040674975629393224403923, −7.84667680771227923452414623968, −7.38691820930490992840648735230, −6.19810416456475013561886774745, −5.84213087499922907811070005687, −4.94314883650609559124876878215, −3.63532242155744052867296199761, −2.58377513223097753871348401518, −1.04109045073573287789850090334, 0.899025128278616773380753080078, 1.76787130691627927487316840522, 2.98048873326306024690373165462, 4.27352586600873827470183411937, 5.23394164653959110980004926051, 6.03286587829133122580757393863, 6.98737369172027260602232437211, 7.81782877740661427559745128246, 8.791375601720376044743843741833, 9.227493025379984313318968702309

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