Properties

Label 2-1470-35.27-c1-0-39
Degree $2$
Conductor $1470$
Sign $-0.925 - 0.378i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.925 - 0.378i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.925 - 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1386217028\)
\(L(\frac12)\) \(\approx\) \(0.1386217028\)
\(L(\frac{3}{2})\) 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

−8.996449761736337497130879250275, −8.438379956767564269489099598940, −7.85904211003159614839143099838, −6.79045169184473963358948514827, −5.53148723441909341649519668926, −4.64251471139207609541391336669, −3.89758836099578182421450855807, −2.74297349712344090262764238963, −1.66678970371135393263578258134, −0.05811793773772736151354023646, 1.83425534472298242389152163744, 2.84394170236271302641391484750, 3.88259481332010388989377564983, 5.16245872423011260811563208424, 6.26176753196388320122660504128, 6.71759176832934356472460780643, 7.69734205579460001320521692601, 8.187382319304747166729204159592, 8.900132496480282363809283908701, 10.14073231192835204073120163295

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