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Properties

Label 2-1734-17.4-c1-0-7
Degree $2$
Conductor $1734$
Sign $-0.914 - 0.405i$
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 + (0.458 − 0.458i)5-s + (−0.707 − 0.707i)6-s + (0.414 + 0.414i)7-s − i·8-s − 1.00i·9-s + (0.458 + 0.458i)10-s + (−2.49 − 2.49i)11-s + (0.707 − 0.707i)12-s + 1.19·13-s + (−0.414 + 0.414i)14-s + 0.648i·15-s + 16-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (0.205 − 0.205i)5-s + (−0.288 − 0.288i)6-s + (0.156 + 0.156i)7-s − 0.353i·8-s − 0.333i·9-s + (0.145 + 0.145i)10-s + (−0.752 − 0.752i)11-s + (0.204 − 0.204i)12-s + 0.332·13-s + (−0.110 + 0.110i)14-s + 0.167i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $-0.914 - 0.405i$
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: \((2,\ 1734,\ (\ :1/2),\ -0.914 - 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9584582298\)
\(L(\frac12)\) \(\approx\) \(0.9584582298\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.458 + 0.458i)T - 5iT^{2} \)
7 \( 1 + (-0.414 - 0.414i)T + 7iT^{2} \)
11 \( 1 + (2.49 + 2.49i)T + 11iT^{2} \)
13 \( 1 - 1.19T + 13T^{2} \)
19 \( 1 - 2.77iT - 19T^{2} \)
23 \( 1 + (-2.19 - 2.19i)T + 23iT^{2} \)
29 \( 1 + (-2.33 + 2.33i)T - 29iT^{2} \)
31 \( 1 + (3.82 - 3.82i)T - 31iT^{2} \)
37 \( 1 + (8.03 - 8.03i)T - 37iT^{2} \)
41 \( 1 + (2.36 + 2.36i)T + 41iT^{2} \)
43 \( 1 - 7.46iT - 43T^{2} \)
47 \( 1 - 9.13T + 47T^{2} \)
53 \( 1 - 12.9iT - 53T^{2} \)
59 \( 1 - 10.4iT - 59T^{2} \)
61 \( 1 + (4.47 + 4.47i)T + 61iT^{2} \)
67 \( 1 - 6.19T + 67T^{2} \)
71 \( 1 + (4.62 - 4.62i)T - 71iT^{2} \)
73 \( 1 + (-8.02 + 8.02i)T - 73iT^{2} \)
79 \( 1 + (0.0846 + 0.0846i)T + 79iT^{2} \)
83 \( 1 - 8.73iT - 83T^{2} \)
89 \( 1 - 1.21T + 89T^{2} \)
97 \( 1 + (9.84 - 9.84i)T - 97iT^{2} \)
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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

−9.512142013971991197073811770752, −8.798354724003308176131478193755, −8.133349357586395683900622584757, −7.25344265961454082007235449457, −6.32644305584390334772721984120, −5.50428051834982896602488306356, −5.10819322770189615822475641805, −3.94911595267418191847131473961, −3.00808381652038882853036335698, −1.29226695299841537362575258443, 0.40589272294363697501154616563, 1.86175493074233579017788579410, 2.64860203517705300749194335317, 3.88880694240836283892884098831, 4.87561479719733273107187607123, 5.55124464407955062687216112729, 6.66728995049931138433993634979, 7.33053805842096260148624432480, 8.254636226882149700777446567116, 9.049383702666842373667106962257

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