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Properties

Label 2-1734-17.4-c1-0-25
Degree $2$
Conductor $1734$
Sign $0.638 + 0.769i$
Analytic cond. $13.8460$
Root an. cond. $3.72102$
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  − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (2.82 − 2.82i)5-s + (0.707 + 0.707i)6-s + (1.41 + 1.41i)7-s + i·8-s − 1.00i·9-s + (−2.82 − 2.82i)10-s + (0.707 − 0.707i)12-s + 6·13-s + (1.41 − 1.41i)14-s + 4.00i·15-s + 16-s − 1.00·18-s + 4i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (1.26 − 1.26i)5-s + (0.288 + 0.288i)6-s + (0.534 + 0.534i)7-s + 0.353i·8-s − 0.333i·9-s + (−0.894 − 0.894i)10-s + (0.204 − 0.204i)12-s + 1.66·13-s + (0.377 − 0.377i)14-s + 1.03i·15-s + 0.250·16-s − 0.235·18-s + 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(13.8460\)
Root analytic conductor: \(3.72102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1734} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1734,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.118779878\)
\(L(\frac12)\) \(\approx\) \(2.118779878\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 \)
good5 \( 1 + (-2.82 + 2.82i)T - 5iT^{2} \)
7 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
11 \( 1 + 11iT^{2} \)
13 \( 1 - 6T + 13T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \)
29 \( 1 + (-2.82 + 2.82i)T - 29iT^{2} \)
31 \( 1 + (4.24 - 4.24i)T - 31iT^{2} \)
37 \( 1 + (2.82 - 2.82i)T - 37iT^{2} \)
41 \( 1 + (-7.07 - 7.07i)T + 41iT^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 + (-2.82 - 2.82i)T + 61iT^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (4.24 - 4.24i)T - 71iT^{2} \)
73 \( 1 + (1.41 - 1.41i)T - 73iT^{2} \)
79 \( 1 + (-7.07 - 7.07i)T + 79iT^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (4.24 - 4.24i)T - 97iT^{2} \)
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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

−9.284296117231742558428266471757, −8.678749113935464235375355975593, −8.092885864477195071309509803214, −6.39696254576639840421097014132, −5.69248826812880945939162036030, −5.15822150412737318690895925776, −4.32562108031942184072451296094, −3.21448737995741067852110916425, −1.74716469390857506638793077964, −1.18716447005332026001439736456, 1.11855930563323365891349631285, 2.34655988128672462293583320183, 3.51693811816339145109280867160, 4.71968002831761964485574728916, 5.76276538350345169889763579322, 6.20650363944950140985794417729, 6.99979779680319481542775363663, 7.45864331393173838062733459929, 8.686502994507477111413620248978, 9.234046011541761291967308079457

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