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Properties

Label 2-735-105.2-c1-0-14
Degree $2$
Conductor $735$
Sign $0.786 + 0.617i$
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  + (−0.455 − 1.69i)2-s + (−1.24 − 1.20i)3-s + (−0.948 + 0.547i)4-s + (0.445 + 2.19i)5-s + (−1.48 + 2.66i)6-s + (−1.12 − 1.12i)8-s + (0.0946 + 2.99i)9-s + (3.52 − 1.75i)10-s + (1.34 − 0.776i)11-s + (1.84 + 0.462i)12-s + (−4.50 + 4.50i)13-s + (2.08 − 3.26i)15-s + (−2.49 + 4.32i)16-s + (2.91 + 0.780i)17-s + (5.05 − 1.52i)18-s + (3.64 + 2.10i)19-s + ⋯
L(s)  = 1  + (−0.322 − 1.20i)2-s + (−0.718 − 0.695i)3-s + (−0.474 + 0.273i)4-s + (0.199 + 0.979i)5-s + (−0.604 + 1.08i)6-s + (−0.397 − 0.397i)8-s + (0.0315 + 0.999i)9-s + (1.11 − 0.554i)10-s + (0.405 − 0.234i)11-s + (0.531 + 0.133i)12-s + (−1.25 + 1.25i)13-s + (0.539 − 0.842i)15-s + (−0.623 + 1.08i)16-s + (0.706 + 0.189i)17-s + (1.19 − 0.359i)18-s + (0.836 + 0.482i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869320 - 0.300694i\)
\(L(\frac12)\) \(\approx\) \(0.869320 - 0.300694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.24 + 1.20i)T \)
5 \( 1 + (-0.445 - 2.19i)T \)
7 \( 1 \)
good2 \( 1 + (0.455 + 1.69i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.34 + 0.776i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.50 - 4.50i)T - 13iT^{2} \)
17 \( 1 + (-2.91 - 0.780i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.64 - 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.13 + 1.37i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
31 \( 1 + (-2.89 - 5.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.68 + 0.450i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.68iT - 41T^{2} \)
43 \( 1 + (2.09 - 2.09i)T - 43iT^{2} \)
47 \( 1 + (0.0131 + 0.0489i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.57 + 5.88i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.46 - 4.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.65 - 2.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.625 + 2.33i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.73iT - 71T^{2} \)
73 \( 1 + (9.92 + 2.66i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.11 - 1.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.2 + 12.2i)T + 83iT^{2} \)
89 \( 1 + (-0.678 + 1.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 10.9i)T + 97iT^{2} \)
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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

−10.29134734662081332638753618578, −9.927943847833923596288495495201, −8.848978459361753961282544286307, −7.46834464902503070466691989627, −6.81088079749336994859903708971, −6.04879231161727382095991149324, −4.75388433624210918721988466774, −3.29699041829446900410769536889, −2.34160451393328864476112921242, −1.24852393675889911372844791351, 0.65427704147802185688747749160, 2.99251609179673706260921886096, 4.62267741363610456777602918693, 5.27885511373910813620955610373, 5.85734132862194320763160545591, 7.02831915807723036550182941103, 7.76192521151806274578222747007, 8.765552111290405677873349879822, 9.559090671599950573062809370766, 10.06176348496255875254217304452

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