↑

Properties

Label 2-930-31.2-c1-0-16
Degree $2$
Conductor $930$
Sign $0.800 + 0.599i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + (0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s − 5-s − 6-s + (−0.186 − 0.573i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (−1.99 − 6.15i)11-s + (−0.809 − 0.587i)12-s + (0.974 − 0.708i)13-s + (0.186 − 0.573i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (1.33 − 4.11i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.154 + 0.475i)4-s − 0.447·5-s − 0.408·6-s + (−0.0703 − 0.216i)7-s + (−0.109 + 0.336i)8-s + (0.103 − 0.317i)9-s + (−0.255 − 0.185i)10-s + (−0.602 − 1.85i)11-s + (−0.233 − 0.169i)12-s + (0.270 − 0.196i)13-s + (0.0497 − 0.153i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.324 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.800 + 0.599i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.800 + 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29568 - 0.431283i\)
\(L(\frac12)\) \(\approx\) \(1.29568 - 0.431283i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + T \)
31 \( 1 + (-1.72 - 5.29i)T \)
good7 \( 1 + (0.186 + 0.573i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.99 + 6.15i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.974 + 0.708i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.33 + 4.11i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.153 - 0.111i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.56 + 4.81i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-6.79 - 4.93i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 + (2.61 + 1.89i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (9.45 + 6.86i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-8.10 + 5.88i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.00 - 3.08i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.96 + 5.06i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 0.666T + 61T^{2} \)
67 \( 1 + 5.98T + 67T^{2} \)
71 \( 1 + (0.550 - 1.69i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.94 + 5.98i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.63 + 8.10i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.585 - 0.425i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.97 + 15.3i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.10 - 12.6i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.42686211568424206777096959735, −8.739099675508344533990940017404, −8.471366188052098354209272071724, −7.21821895283707455911274281532, −6.50896886617966386048919820071, −5.45250549614384748448942709404, −4.91601910637468500103534704809, −3.61211227028479354174752588806, −2.97610683957910164950265735923, −0.58953110164667405379417074222, 1.49962371620769368390457206544, 2.63174345676919822035924269928, 4.01043664223901737804001865937, 4.77697083781511193066338464371, 5.71074690682348926925656708204, 6.68448287193657773729329010617, 7.48419872872704195243721597551, 8.337724251151585689347249923941, 9.676539633345882427774493972245, 10.20717252443470533361909468917

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