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Properties

Label 2-735-105.53-c1-0-31
Degree $2$
Conductor $735$
Sign $0.628 - 0.778i$
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  + (0.127 − 0.474i)2-s + (1.58 + 0.708i)3-s + (1.52 + 0.879i)4-s + (−1.06 + 1.96i)5-s + (0.536 − 0.659i)6-s + (1.30 − 1.30i)8-s + (1.99 + 2.24i)9-s + (0.795 + 0.755i)10-s + (2.31 + 1.33i)11-s + (1.78 + 2.46i)12-s + (−2.14 − 2.14i)13-s + (−3.07 + 2.34i)15-s + (1.30 + 2.26i)16-s + (−4.46 + 1.19i)17-s + (1.31 − 0.661i)18-s + (4.54 − 2.62i)19-s + ⋯
L(s)  = 1  + (0.0898 − 0.335i)2-s + (0.912 + 0.409i)3-s + (0.761 + 0.439i)4-s + (−0.477 + 0.878i)5-s + (0.219 − 0.269i)6-s + (0.461 − 0.461i)8-s + (0.665 + 0.746i)9-s + (0.251 + 0.238i)10-s + (0.697 + 0.402i)11-s + (0.515 + 0.712i)12-s + (−0.596 − 0.596i)13-s + (−0.795 + 0.606i)15-s + (0.326 + 0.565i)16-s + (−1.08 + 0.290i)17-s + (0.310 − 0.155i)18-s + (1.04 − 0.601i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.628 - 0.778i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.628 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21811 + 1.06012i\)
\(L(\frac12)\) \(\approx\) \(2.21811 + 1.06012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 - 0.708i)T \)
5 \( 1 + (1.06 - 1.96i)T \)
7 \( 1 \)
good2 \( 1 + (-0.127 + 0.474i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-2.31 - 1.33i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.14 + 2.14i)T + 13iT^{2} \)
17 \( 1 + (4.46 - 1.19i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.54 + 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.48 + 0.932i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.86T + 29T^{2} \)
31 \( 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.92 - 0.784i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (-0.759 - 0.759i)T + 43iT^{2} \)
47 \( 1 + (2.80 - 10.4i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.62 + 6.05i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.0797 - 0.138i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.36 - 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.98 + 7.39i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (-5.68 + 1.52i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.37 - 1.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.03 + 4.03i)T - 83iT^{2} \)
89 \( 1 + (1.97 + 3.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.86 + 1.86i)T - 97iT^{2} \)
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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

−10.55230189509729335305647757917, −9.803206978510878870997088104807, −8.792454498842436720453762189541, −7.75014910261249361194108812812, −7.23894018924505071261624085332, −6.39424577858755610819596007155, −4.67828927106037726899802445870, −3.71729281310738522385676697097, −2.93236893220651445807558207802, −2.02632625316240747212084670365, 1.22689464464737416397871378944, 2.35021529137552103239788438120, 3.74687118788334083503696456647, 4.76146271593818886539002940357, 5.98977113317056953122564410631, 6.89752142218393925000407295749, 7.65464509549854232511467637066, 8.407128214495595795550594704780, 9.336041604013607484731276929788, 9.953553059744547397775748405855

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