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Properties

Label 2-1470-35.27-c1-0-5
Degree $2$
Conductor $1470$
Sign $-0.333 - 0.942i$
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 + (0.461 + 2.18i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (1.22 − 1.87i)10-s − 2.98·11-s + (0.707 − 0.707i)12-s + (−0.960 − 0.960i)13-s + (1.22 − 1.87i)15-s − 1.00·16-s + (−1.62 + 1.62i)17-s + (0.707 − 0.707i)18-s + 8.67·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.206 + 0.978i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.385 − 0.592i)10-s − 0.899·11-s + (0.204 − 0.204i)12-s + (−0.266 − 0.266i)13-s + (0.315 − 0.483i)15-s − 0.250·16-s + (−0.394 + 0.394i)17-s + (0.166 − 0.166i)18-s + 1.98·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.333 - 0.942i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.333 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4749893396\)
\(L(\frac12)\) \(\approx\) \(0.4749893396\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.461 - 2.18i)T \)
7 \( 1 \)
good11 \( 1 + 2.98T + 11T^{2} \)
13 \( 1 + (0.960 + 0.960i)T + 13iT^{2} \)
17 \( 1 + (1.62 - 1.62i)T - 17iT^{2} \)
19 \( 1 - 8.67T + 19T^{2} \)
23 \( 1 + (-1.36 + 1.36i)T - 23iT^{2} \)
29 \( 1 - 2.00iT - 29T^{2} \)
31 \( 1 - 0.179iT - 31T^{2} \)
37 \( 1 + (4.86 + 4.86i)T + 37iT^{2} \)
41 \( 1 - 5.14iT - 41T^{2} \)
43 \( 1 + (7.01 - 7.01i)T - 43iT^{2} \)
47 \( 1 + (-0.202 + 0.202i)T - 47iT^{2} \)
53 \( 1 + (7.01 - 7.01i)T - 53iT^{2} \)
59 \( 1 + 7.09T + 59T^{2} \)
61 \( 1 - 2.41iT - 61T^{2} \)
67 \( 1 + (6.29 + 6.29i)T + 67iT^{2} \)
71 \( 1 + 9.08T + 71T^{2} \)
73 \( 1 + (8.78 + 8.78i)T + 73iT^{2} \)
79 \( 1 - 16.2iT - 79T^{2} \)
83 \( 1 + (-8.11 - 8.11i)T + 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + (-7.44 + 7.44i)T - 97iT^{2} \)
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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

−9.916205598794640671942418053740, −9.123515916078323181395069198016, −7.917641887110297045947388214100, −7.49242974771367992072408173405, −6.65409428101839260309547922167, −5.71400716031413787959689353109, −4.81014766791964966200606753208, −3.31960584419453372403274072222, −2.68370379698863319864645552367, −1.44891190248953267679651567613, 0.24092409628550973267663530484, 1.62334109794843661474721453982, 3.15782824628179088702868965640, 4.57291448363719071077750085292, 5.21066512370421166720321395060, 5.75812053830462583663203862397, 6.95703160331051175072458271695, 7.67246945486024083120526500143, 8.560063910631368598060637591167, 9.290043716010047046205633037558

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