↑

Properties

Label 2-735-105.2-c1-0-33
Degree $2$
Conductor $735$
Sign $0.680 - 0.732i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.680 - 0.732i$
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: \((2,\ 735,\ (\ :1/2),\ 0.680 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58379 + 0.690679i\)
\(L(\frac12)\) \(\approx\) \(1.58379 + 0.690679i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 1.86i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.47384182224864309983188896842, −10.10081972930627962966677533833, −8.879705675830054778591667353561, −7.56613841543770518192211471056, −6.69768786899558493119542920937, −6.01545129425335112727522854260, −5.56766868391930248927463603507, −4.13460992362046121578247564871, −2.89632053169696257967243217090, −1.32402751720145317807885932688, 1.23953358025521541984553933315, 2.12603784901940107726039177702, 3.89042656596756624569030141441, 4.77168513327494063285843412324, 6.13081601864346734183609079163, 6.42309229790793200203478852202, 7.66309079184508406286922228787, 8.394417272395742836375189912818, 9.710664057901884179170644475043, 10.31246756101865298687501548983

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