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Properties

Label 2-1470-35.27-c1-0-31
Degree $2$
Conductor $1470$
Sign $0.967 + 0.251i$
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 + (1.35 − 1.77i)5-s − 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (2.21 − 0.300i)10-s + 2.44·11-s + (0.707 − 0.707i)12-s + (1.28 + 1.28i)13-s + (−2.21 + 0.300i)15-s − 1.00·16-s + (5.33 − 5.33i)17-s + (−0.707 + 0.707i)18-s − 0.690·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.605 − 0.795i)5-s − 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.700 − 0.0951i)10-s + 0.738·11-s + (0.204 − 0.204i)12-s + (0.356 + 0.356i)13-s + (−0.572 + 0.0776i)15-s − 0.250·16-s + (1.29 − 1.29i)17-s + (−0.166 + 0.166i)18-s − 0.158·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.967 + 0.251i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.274030686\)
\(L(\frac12)\) \(\approx\) \(2.274030686\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 + 1.77i)T \)
7 \( 1 \)
good11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + (-1.28 - 1.28i)T + 13iT^{2} \)
17 \( 1 + (-5.33 + 5.33i)T - 17iT^{2} \)
19 \( 1 + 0.690T + 19T^{2} \)
23 \( 1 + (5.65 - 5.65i)T - 23iT^{2} \)
29 \( 1 + 5.08iT - 29T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + (-7.16 - 7.16i)T + 37iT^{2} \)
41 \( 1 + 1.15iT - 41T^{2} \)
43 \( 1 + (-0.893 + 0.893i)T - 43iT^{2} \)
47 \( 1 + (-4.95 + 4.95i)T - 47iT^{2} \)
53 \( 1 + (-6.74 + 6.74i)T - 53iT^{2} \)
59 \( 1 - 8.06T + 59T^{2} \)
61 \( 1 + 1.09iT - 61T^{2} \)
67 \( 1 + (7.41 + 7.41i)T + 67iT^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + (-4.58 - 4.58i)T + 73iT^{2} \)
79 \( 1 + 2.31iT - 79T^{2} \)
83 \( 1 + (-7.39 - 7.39i)T + 83iT^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + (-4.74 + 4.74i)T - 97iT^{2} \)
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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

−9.559442970417338899954178009308, −8.512523969331465367953727412052, −7.77836448312844520449438799163, −6.92090748932613708617447707980, −6.00200613094245250680757711152, −5.53742336826783667091048655212, −4.60021610303705217100421499466, −3.65853768956237294594946689254, −2.20893069460980058215912206166, −0.964106221508353700798454926005, 1.27600883399244654763479943101, 2.52707143991874158810770328614, 3.61500059975137253696921644657, 4.24217489416657404571875902975, 5.61964378425238742203021024042, 5.98267565355312748010847936793, 6.79247208819712250517758687141, 7.935253631635901739055667104968, 8.997813818359225215859194268124, 9.819646435801254310862647204136

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