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Properties

Label 2-1734-17.13-c1-0-22
Degree $2$
Conductor $1734$
Sign $-0.122 + 0.992i$
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 + (−0.414 − 0.414i)5-s + (−0.707 + 0.707i)6-s + (−3.41 + 3.41i)7-s + i·8-s + 1.00i·9-s + (−0.414 + 0.414i)10-s + (2.82 − 2.82i)11-s + (0.707 + 0.707i)12-s − 2.82·13-s + (3.41 + 3.41i)14-s + 0.585i·15-s + 16-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.185 − 0.185i)5-s + (−0.288 + 0.288i)6-s + (−1.29 + 1.29i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.130 + 0.130i)10-s + (0.852 − 0.852i)11-s + (0.204 + 0.204i)12-s − 0.784·13-s + (0.912 + 0.912i)14-s + 0.151i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(13.8460\)
Root analytic conductor: \(3.72102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1734} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1734,\ (\ :1/2),\ -0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9088879749\)
\(L(\frac12)\) \(\approx\) \(0.9088879749\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.414 + 0.414i)T + 5iT^{2} \)
7 \( 1 + (3.41 - 3.41i)T - 7iT^{2} \)
11 \( 1 + (-2.82 + 2.82i)T - 11iT^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
19 \( 1 - 6.82iT - 19T^{2} \)
23 \( 1 + (-2.24 + 2.24i)T - 23iT^{2} \)
29 \( 1 + (-0.414 - 0.414i)T + 29iT^{2} \)
31 \( 1 + (0.585 + 0.585i)T + 31iT^{2} \)
37 \( 1 + (-1.58 - 1.58i)T + 37iT^{2} \)
41 \( 1 + (-2.17 + 2.17i)T - 41iT^{2} \)
43 \( 1 - 1.65iT - 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 2.82iT - 53T^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 + (0.757 - 0.757i)T - 61iT^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + (4.58 + 4.58i)T + 71iT^{2} \)
73 \( 1 + (6.65 + 6.65i)T + 73iT^{2} \)
79 \( 1 + (-10.2 + 10.2i)T - 79iT^{2} \)
83 \( 1 + 4.48iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-9.48 - 9.48i)T + 97iT^{2} \)
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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

−9.116864393645726413804421713405, −8.589637807241450183277455844627, −7.60488641380437375455254232975, −6.33401640303732980435004047703, −6.04944810810179606470722430549, −5.05074358004740930660152782121, −3.85250334087151456132074307472, −2.98815700786283812740402780319, −2.03573216501012387839094192243, −0.52274581596858447651493779110, 0.833846377712024398168587328385, 2.89681549321638054988516613923, 3.96528573393230797957663179405, 4.48332779488844833449624175996, 5.53239807762568446308904535717, 6.58143728929926625533055443284, 7.14802580432125113899065777748, 7.42863705843390222819579607103, 9.086178119846886109208775205593, 9.392120719809480762591145916761

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