↑

Properties

Label 2-735-105.32-c1-0-69
Degree $2$
Conductor $735$
Sign $-0.262 + 0.965i$
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  + (1.46 + 0.391i)2-s + (0.852 − 1.50i)3-s + (0.246 + 0.142i)4-s + (−0.207 − 2.22i)5-s + (1.83 − 1.86i)6-s + (−1.83 − 1.83i)8-s + (−1.54 − 2.57i)9-s + (0.567 − 3.33i)10-s + (0.791 + 0.457i)11-s + (0.425 − 0.250i)12-s + (−3.07 + 3.07i)13-s + (−3.53 − 1.58i)15-s + (−2.24 − 3.88i)16-s + (0.311 + 1.16i)17-s + (−1.24 − 4.35i)18-s + (5.95 − 3.43i)19-s + ⋯
L(s)  = 1  + (1.03 + 0.276i)2-s + (0.492 − 0.870i)3-s + (0.123 + 0.0712i)4-s + (−0.0929 − 0.995i)5-s + (0.749 − 0.762i)6-s + (−0.648 − 0.648i)8-s + (−0.514 − 0.857i)9-s + (0.179 − 1.05i)10-s + (0.238 + 0.137i)11-s + (0.122 − 0.0723i)12-s + (−0.854 + 0.854i)13-s + (−0.912 − 0.409i)15-s + (−0.561 − 0.971i)16-s + (0.0755 + 0.281i)17-s + (−0.294 − 1.02i)18-s + (1.36 − 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.262 + 0.965i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.262 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46587 - 1.91689i\)
\(L(\frac12)\) \(\approx\) \(1.46587 - 1.91689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.852 + 1.50i)T \)
5 \( 1 + (0.207 + 2.22i)T \)
7 \( 1 \)
good2 \( 1 + (-1.46 - 0.391i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-0.791 - 0.457i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.07 - 3.07i)T - 13iT^{2} \)
17 \( 1 + (-0.311 - 1.16i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.95 + 3.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.505 + 1.88i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.72T + 29T^{2} \)
31 \( 1 + (-2.31 + 4.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.207 + 0.774i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 - 4.80i)T - 43iT^{2} \)
47 \( 1 + (-10.1 - 2.71i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-10.6 + 2.85i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.94 + 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.533 + 0.924i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.83 + 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.557iT - 71T^{2} \)
73 \( 1 + (-0.564 - 2.10i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.62 - 1.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.38 - 2.38i)T + 83iT^{2} \)
89 \( 1 + (-5.64 - 9.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \)
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

−9.717748823349102460231899778468, −9.270493973033579927611176930656, −8.360909650249607807121191534330, −7.31875988274223338485538173875, −6.58284877065702073843357391384, −5.53305438860969618370449024516, −4.70451451338448288798908710973, −3.76851416708513057626236748178, −2.45384056247865419591204769802, −0.866774209962843360405339723921, 2.55229061345432362210517692684, 3.22910282597295126955578554438, 3.98253395073092015775515899015, 5.14198469212468957830193753748, 5.74139417652411879066640131076, 7.17301043447075093990784038499, 8.038711750805072746955053586983, 9.033782732614842245292014693422, 10.03731658705636054958129319401, 10.52133822364349202954076725251

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