↑

Properties

Label 2-735-105.59-c1-0-11
Degree $2$
Conductor $735$
Sign $-0.205 - 0.978i$
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  + (0.866 + 1.5i)2-s + (−0.724 − 1.57i)3-s + (−0.5 + 0.866i)4-s + (−0.358 + 2.20i)5-s + (1.73 − 2.44i)6-s + 1.73·8-s + (−1.94 + 2.28i)9-s + (−3.62 + 1.37i)10-s + (2.44 + 1.41i)11-s + (1.72 + 0.158i)12-s − 4·13-s + (3.73 − 1.03i)15-s + (2.49 + 4.33i)16-s + (2.44 + 1.41i)17-s + (−5.10 − 0.949i)18-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (−0.418 − 0.908i)3-s + (−0.250 + 0.433i)4-s + (−0.160 + 0.987i)5-s + (0.707 − 0.999i)6-s + 0.612·8-s + (−0.649 + 0.760i)9-s + (−1.14 + 0.434i)10-s + (0.738 + 0.426i)11-s + (0.497 + 0.0458i)12-s − 1.10·13-s + (0.963 − 0.267i)15-s + (0.624 + 1.08i)16-s + (0.594 + 0.342i)17-s + (−1.20 − 0.223i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.205 - 0.978i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13004 + 1.39232i\)
\(L(\frac12)\) \(\approx\) \(1.13004 + 1.39232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.724 + 1.57i)T \)
5 \( 1 + (0.358 - 2.20i)T \)
7 \( 1 \)
good2 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-2.44 - 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-2.44 - 1.41i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 + (-8.48 - 4.89i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 4.89iT - 43T^{2} \)
47 \( 1 + (2.44 - 1.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.48 + 4.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.24 + 2.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + (4 - 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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

−10.64897133933805750061884821166, −9.945813472596413727418652479001, −8.467492967423846487302543800823, −7.46470393134625206600392961367, −7.05739800542809466036136446940, −6.40239106758061233516824084650, −5.50703271376223692838226585253, −4.58093588597433886873497554546, −3.13590258561423064632481799696, −1.67171551042371605820102132597, 0.853164265812425134483059573354, 2.58685623545235453441989597160, 3.73302104843163290913015596991, 4.54199410894471429019679145461, 5.11479587945157692069868481757, 6.26389601881829198627199646886, 7.69550288779461427135744839938, 8.681633442117437220734689236342, 9.737878266216857318331456977321, 10.06588960996230331958243240812

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