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Properties

Label 2-1470-35.27-c1-0-33
Degree $2$
Conductor $1470$
Sign $-0.563 + 0.826i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.563 + 0.826i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.563 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4197167263\)
\(L(\frac12)\) \(\approx\) \(0.4197167263\)
\(L(\frac{3}{2})\) 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

−9.397631840080309395531934706454, −8.289650580984055993457168499375, −7.934797258877830542124558923442, −7.03091764344298912706347164769, −6.07340479572662674579381101323, −4.69585429611778725986130013418, −3.93698110907697947733130878959, −2.94954916672459536501612588700, −2.21257053954957623074654664590, −0.19148171509893983093684336712, 1.31828585875633489871973575531, 2.53707285911341548800854492596, 3.98922482002520817759918123039, 4.75106550006008382126123324374, 5.78959616143872145089996818674, 6.85740081880260091306570659700, 7.38199311579027204038399874352, 8.336145240358795248844763938754, 8.724739019126816432668194545361, 9.494435362295286573520222272956

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