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Properties

Label 2-1734-17.4-c1-0-16
Degree $2$
Conductor $1734$
Sign $0.999 + 0.0419i$
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 + (−0.892 + 0.892i)5-s + (−0.707 − 0.707i)6-s + (2.08 + 2.08i)7-s + i·8-s − 1.00i·9-s + (0.892 + 0.892i)10-s + (−2.33 − 2.33i)11-s + (−0.707 + 0.707i)12-s + 3.36·13-s + (2.08 − 2.08i)14-s + 1.26i·15-s + 16-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (−0.399 + 0.399i)5-s + (−0.288 − 0.288i)6-s + (0.787 + 0.787i)7-s + 0.353i·8-s − 0.333i·9-s + (0.282 + 0.282i)10-s + (−0.703 − 0.703i)11-s + (−0.204 + 0.204i)12-s + 0.932·13-s + (0.556 − 0.556i)14-s + 0.325i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $0.999 + 0.0419i$
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: \((2,\ 1734,\ (\ :1/2),\ 0.999 + 0.0419i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.765143278\)
\(L(\frac12)\) \(\approx\) \(1.765143278\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.892 - 0.892i)T - 5iT^{2} \)
7 \( 1 + (-2.08 - 2.08i)T + 7iT^{2} \)
11 \( 1 + (2.33 + 2.33i)T + 11iT^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
19 \( 1 - 7.44iT - 19T^{2} \)
23 \( 1 + (-1 - i)T + 23iT^{2} \)
29 \( 1 + (-0.369 + 0.369i)T - 29iT^{2} \)
31 \( 1 + (5.02 - 5.02i)T - 31iT^{2} \)
37 \( 1 + (-0.560 + 0.560i)T - 37iT^{2} \)
41 \( 1 + (-2.87 - 2.87i)T + 41iT^{2} \)
43 \( 1 + 0.867iT - 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 9.98iT - 53T^{2} \)
59 \( 1 - 7.31iT - 59T^{2} \)
61 \( 1 + (-7.56 - 7.56i)T + 61iT^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + (5.21 - 5.21i)T - 71iT^{2} \)
73 \( 1 + (3.70 - 3.70i)T - 73iT^{2} \)
79 \( 1 + (7.73 + 7.73i)T + 79iT^{2} \)
83 \( 1 - 12.3iT - 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + (4.40 - 4.40i)T - 97iT^{2} \)
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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

−9.160206183587680275250682798833, −8.451524477204862719917533758475, −8.039694486208384870460497913262, −7.12671826645571683654471054238, −5.83964945220015401792586358220, −5.36166473880716699946384519786, −3.93957220962374062172924555355, −3.29085370019316235167737880154, −2.26287587935036179384709972568, −1.26436760026912838720906519526, 0.71665291691500704432854883382, 2.32856355200721069508060904223, 3.72286887715397827895033180588, 4.52821884919679727554954284545, 4.96911182449692464225359976034, 6.11629220350722313306495123843, 7.28096137418306725263560630392, 7.63109131680001777683027575968, 8.531673450912289052521043487717, 9.039027356331444579523786911958

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