↑

Properties

Label 2-630-63.5-c1-0-29
Degree $2$
Conductor $630$
Sign $-0.943 + 0.331i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
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  − i·2-s + (1.70 − 0.296i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−0.296 − 1.70i)6-s + (−2.21 − 1.44i)7-s + i·8-s + (2.82 − 1.01i)9-s + (−0.866 + 0.5i)10-s + (−4.20 − 2.43i)11-s + (−1.70 + 0.296i)12-s + (−3.42 − 1.97i)13-s + (−1.44 + 2.21i)14-s + (−1.11 − 1.32i)15-s + 16-s + (1.25 + 2.16i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.985 − 0.171i)3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.121 − 0.696i)6-s + (−0.837 − 0.545i)7-s + 0.353i·8-s + (0.941 − 0.337i)9-s + (−0.273 + 0.158i)10-s + (−1.26 − 0.732i)11-s + (−0.492 + 0.0856i)12-s + (−0.948 − 0.547i)13-s + (−0.385 + 0.592i)14-s + (−0.286 − 0.343i)15-s + 0.250·16-s + (0.303 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.943 + 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.215217 - 1.25998i\)
\(L(\frac12)\) \(\approx\) \(0.215217 - 1.25998i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.70 + 0.296i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.21 + 1.44i)T \)
good11 \( 1 + (4.20 + 2.43i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.42 + 1.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.25 - 2.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.962 + 0.555i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.86 + 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.70 + 2.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.39iT - 31T^{2} \)
37 \( 1 + (1.82 - 3.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.90 - 6.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.62 + 6.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + (-4.92 + 2.84i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.39T + 59T^{2} \)
61 \( 1 + 9.20iT - 61T^{2} \)
67 \( 1 - 6.01T + 67T^{2} \)
71 \( 1 - 9.66iT - 71T^{2} \)
73 \( 1 + (-12.0 + 6.96i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.83T + 79T^{2} \)
83 \( 1 + (-0.393 - 0.682i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.49 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.9 - 8.64i)T + (48.5 - 84.0i)T^{2} \)
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

−10.13019034579477086554069021176, −9.522811634515251449992375912327, −8.390350144295721939876927550185, −7.909627781999928364127676647265, −6.83512137333715033438619498164, −5.42700394608630247551121548205, −4.26245528537761007673806081640, −3.23570678344619202033483675226, −2.45318488842799980406820454163, −0.59909570611193917411908643241, 2.38442872755585621923386169617, 3.25060267331672462722233610382, 4.57923443724909943707426740723, 5.45851809786697134775661474636, 7.03422306559214999740453283149, 7.21284108007867968598532909545, 8.362454232593583638179929249892, 9.158704035462156414524431112181, 9.933130737046214771378155305789, 10.54794022087189087090925903513

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