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Properties

Label 2-1470-35.13-c1-0-6
Degree $2$
Conductor $1470$
Sign $-0.696 - 0.717i$
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 + (−1.99 + 1.00i)5-s − 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (0.700 − 2.12i)10-s − 3.10·11-s + (0.707 + 0.707i)12-s + (3.40 − 3.40i)13-s + (0.700 − 2.12i)15-s − 1.00·16-s + (3.76 + 3.76i)17-s + (0.707 + 0.707i)18-s + 7.23·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (−0.892 + 0.450i)5-s − 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.221 − 0.671i)10-s − 0.937·11-s + (0.204 + 0.204i)12-s + (0.945 − 0.945i)13-s + (0.180 − 0.548i)15-s − 0.250·16-s + (0.913 + 0.913i)17-s + (0.166 + 0.166i)18-s + 1.65·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.696 - 0.717i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6957494542\)
\(L(\frac12)\) \(\approx\) \(0.6957494542\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.99 - 1.00i)T \)
7 \( 1 \)
good11 \( 1 + 3.10T + 11T^{2} \)
13 \( 1 + (-3.40 + 3.40i)T - 13iT^{2} \)
17 \( 1 + (-3.76 - 3.76i)T + 17iT^{2} \)
19 \( 1 - 7.23T + 19T^{2} \)
23 \( 1 + (3.72 + 3.72i)T + 23iT^{2} \)
29 \( 1 - 4.49iT - 29T^{2} \)
31 \( 1 - 9.22iT - 31T^{2} \)
37 \( 1 + (2.54 - 2.54i)T - 37iT^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 + (3.86 + 3.86i)T + 43iT^{2} \)
47 \( 1 + (2.88 + 2.88i)T + 47iT^{2} \)
53 \( 1 + (-2.43 - 2.43i)T + 53iT^{2} \)
59 \( 1 + 1.33T + 59T^{2} \)
61 \( 1 - 12.5iT - 61T^{2} \)
67 \( 1 + (4.71 - 4.71i)T - 67iT^{2} \)
71 \( 1 - 9.22T + 71T^{2} \)
73 \( 1 + (-5.82 + 5.82i)T - 73iT^{2} \)
79 \( 1 - 1.91iT - 79T^{2} \)
83 \( 1 + (8.97 - 8.97i)T - 83iT^{2} \)
89 \( 1 + 4.07T + 89T^{2} \)
97 \( 1 + (2.69 + 2.69i)T + 97iT^{2} \)
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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

−10.14469813982219780784997808781, −8.740461639941393579788652766083, −8.193589603635616737871071077974, −7.49727983217085272408903831880, −6.65852927514028802380341605116, −5.63392809204835718164061255653, −5.09441285558395628888322161885, −3.74708706640710970626551441651, −3.04373975027200435436247334805, −1.07457246767995609028747281810, 0.43933007904804821566173079180, 1.60528109559985926771155382029, 3.02487396696573286552281691243, 3.94289157863700367797201221535, 5.00889910576429552957952685304, 5.85926371348027230606110861725, 7.08544534541781904271766331479, 7.82237212995236510071031322238, 8.154243750294015056815850358844, 9.432560814920006192775122943290

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