Properties

Label 2-1470-35.27-c1-0-25
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: $\chi_{1470} (97, \cdot )$
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

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

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