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Properties

Label 2-735-105.89-c1-0-27
Degree $2$
Conductor $735$
Sign $0.982 - 0.188i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
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  + (0.701 − 1.21i)2-s + (−1.5 + 0.866i)3-s + (0.0157 + 0.0272i)4-s + (1.93 + 1.11i)5-s + 2.43i·6-s + 2.85·8-s + (1.5 − 2.59i)9-s + (2.71 − 1.56i)10-s + (−0.0472 − 0.0272i)12-s − 3.87·15-s + (1.96 − 3.40i)16-s + (−6.98 + 4.03i)17-s + (−2.10 − 3.64i)18-s + (5.76 + 3.32i)19-s + 0.0704i·20-s + ⋯
L(s)  = 1  + (0.496 − 0.859i)2-s + (−0.866 + 0.499i)3-s + (0.00787 + 0.0136i)4-s + (0.866 + 0.499i)5-s + 0.992i·6-s + 1.00·8-s + (0.5 − 0.866i)9-s + (0.859 − 0.496i)10-s + (−0.0136 − 0.00787i)12-s − 1.00·15-s + (0.491 − 0.852i)16-s + (−1.69 + 0.977i)17-s + (−0.496 − 0.859i)18-s + (1.32 + 0.763i)19-s + 0.0157i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.982 - 0.188i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91895 + 0.182513i\)
\(L(\frac12)\) \(\approx\) \(1.91895 + 0.182513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 \)
good2 \( 1 + (-0.701 + 1.21i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (6.98 - 4.03i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.76 - 3.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.111 - 0.192i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-8.84 + 5.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (0.886 + 0.511i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.54 - 9.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (13.0 + 7.53i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.91 - 5.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.0iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.50703337102210298161302633942, −10.01715317723032320004227329080, −9.059555977474144378979279061003, −7.68037992021194934683674501212, −6.60183819434215832765471560444, −5.88789743199492640759808375927, −4.78602030502439427327725364607, −3.92599101477187146245069125615, −2.80088856083354210935939408992, −1.55599893875830309781945744555, 1.06944182156322602394991522636, 2.41012262124259710933505301600, 4.67238919055432237512579345245, 5.00707427160934197894986710117, 5.97024156465507578616722291308, 6.70864321116137886435947618551, 7.27424949617216490460142953521, 8.487402118520978122412752119478, 9.550011786532002029413263756167, 10.40619543048951888408385927793

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