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Properties

Label 2-1710-15.8-c1-0-21
Degree $2$
Conductor $1710$
Sign $0.222 - 0.974i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
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 + 1.00i·4-s + (1.06 + 1.96i)5-s + (3.58 − 3.58i)7-s + (−0.707 + 0.707i)8-s + (−0.632 + 2.14i)10-s + 4.10i·11-s + (−1.38 − 1.38i)13-s + 5.07·14-s − 1.00·16-s + (1.61 + 1.61i)17-s − i·19-s + (−1.96 + 1.06i)20-s + (−2.90 + 2.90i)22-s + (−4.18 + 4.18i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.478 + 0.878i)5-s + (1.35 − 1.35i)7-s + (−0.250 + 0.250i)8-s + (−0.199 + 0.678i)10-s + 1.23i·11-s + (−0.384 − 0.384i)13-s + 1.35·14-s − 0.250·16-s + (0.392 + 0.392i)17-s − 0.229i·19-s + (−0.439 + 0.239i)20-s + (−0.618 + 0.618i)22-s + (−0.871 + 0.871i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.222 - 0.974i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.222 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.846576618\)
\(L(\frac12)\) \(\approx\) \(2.846576618\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.06 - 1.96i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-3.58 + 3.58i)T - 7iT^{2} \)
11 \( 1 - 4.10iT - 11T^{2} \)
13 \( 1 + (1.38 + 1.38i)T + 13iT^{2} \)
17 \( 1 + (-1.61 - 1.61i)T + 17iT^{2} \)
23 \( 1 + (4.18 - 4.18i)T - 23iT^{2} \)
29 \( 1 - 9.09T + 29T^{2} \)
31 \( 1 - 8.32T + 31T^{2} \)
37 \( 1 + (1.38 - 1.38i)T - 37iT^{2} \)
41 \( 1 + 2.96iT - 41T^{2} \)
43 \( 1 + (-5.42 - 5.42i)T + 43iT^{2} \)
47 \( 1 + (7.09 + 7.09i)T + 47iT^{2} \)
53 \( 1 + (-0.514 + 0.514i)T - 53iT^{2} \)
59 \( 1 + 0.814T + 59T^{2} \)
61 \( 1 + 2.90T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 + 3.04iT - 71T^{2} \)
73 \( 1 + (-6.42 - 6.42i)T + 73iT^{2} \)
79 \( 1 - 10.9iT - 79T^{2} \)
83 \( 1 + (8.82 - 8.82i)T - 83iT^{2} \)
89 \( 1 - 4.88T + 89T^{2} \)
97 \( 1 + (7.03 - 7.03i)T - 97iT^{2} \)
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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

−9.881340923968266931564097358574, −8.284506596743684563557456248432, −7.77001662481805924398317722675, −7.08802687211858789884045071452, −6.47153531584511959753282504502, −5.30416169755039681231658927309, −4.60129397199001450079018016952, −3.83885285607286112846026155759, −2.59271311687310957772829743439, −1.45502944244927230499592771353, 1.02723472786398099532833521522, 2.11593057956916581847892371147, 2.91493297725897178287621318321, 4.48617469223683721909178060323, 4.89712012313644611473146153572, 5.80826754939116132863338477941, 6.27931207175696269187636627020, 7.968860093834553021051021549094, 8.461102968889537741934645228923, 9.059920635649304019140882068181

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