↑

Properties

Label 2-414-23.13-c1-0-6
Degree $2$
Conductor $414$
Sign $0.0635 + 0.997i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (−0.154 + 0.339i)5-s + (−1.97 − 0.580i)7-s + (0.841 − 0.540i)8-s + (0.357 − 0.105i)10-s + (1.61 − 1.86i)11-s + (3.58 − 1.05i)13-s + (0.855 + 1.87i)14-s + (−0.959 − 0.281i)16-s + (−0.897 − 6.24i)17-s + (0.468 − 3.25i)19-s + (−0.313 − 0.201i)20-s − 2.46·22-s + (1.76 − 4.45i)23-s + ⋯
L(s)  = 1  + (−0.463 − 0.534i)2-s + (−0.0711 + 0.494i)4-s + (−0.0692 + 0.151i)5-s + (−0.746 − 0.219i)7-s + (0.297 − 0.191i)8-s + (0.113 − 0.0332i)10-s + (0.486 − 0.561i)11-s + (0.993 − 0.291i)13-s + (0.228 + 0.500i)14-s + (−0.239 − 0.0704i)16-s + (−0.217 − 1.51i)17-s + (0.107 − 0.746i)19-s + (−0.0701 − 0.0450i)20-s − 0.525·22-s + (0.367 − 0.929i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0635 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0635 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.0635 + 0.997i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.0635 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703211 - 0.659881i\)
\(L(\frac12)\) \(\approx\) \(0.703211 - 0.659881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
3 \( 1 \)
23 \( 1 + (-1.76 + 4.45i)T \)
good5 \( 1 + (0.154 - 0.339i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (1.97 + 0.580i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-1.61 + 1.86i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-3.58 + 1.05i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (0.897 + 6.24i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.468 + 3.25i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-1.01 - 7.06i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-4.91 + 3.16i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (1.94 + 4.25i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-1.98 + 4.34i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (9.36 + 6.02i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 2.44T + 47T^{2} \)
53 \( 1 + (-5.80 - 1.70i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (11.3 - 3.32i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-1.41 + 0.911i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-9.23 - 10.6i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-2.64 - 3.05i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.23 - 8.60i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-2.28 + 0.671i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (1.76 + 3.85i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-7.59 - 4.88i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (4.69 - 10.2i)T + (-63.5 - 73.3i)T^{2} \)
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

−11.00787314973494705209009652298, −10.17354147602244197493733451760, −9.105978014552437580169594483161, −8.627950920220882621481220747549, −7.19925944617074208434472265364, −6.54286499370701285671403746243, −5.09333616468274778449553964861, −3.65981647542577602170673505486, −2.78863687939775986160192246394, −0.800458643097550356692580623378, 1.55698977941023028016617507658, 3.46597896291166487650241793002, 4.66988475304425052454088349910, 6.22672198801257141986118002006, 6.45315591842732106630752987343, 7.927098263472876196301065254624, 8.603014767068510452147074799339, 9.603070995665416242736797312357, 10.26225262628823565256461842619, 11.36144052854033690379101655627

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