Properties

Label 2-930-93.68-c1-0-17
Degree $2$
Conductor $930$
Sign $0.699 + 0.714i$
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 about

Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + (−0.474 − 1.66i)3-s − 4-s + (0.866 + 0.5i)5-s + (−1.66 + 0.474i)6-s + (1.75 + 3.03i)7-s + i·8-s + (−2.54 + 1.58i)9-s + (0.5 − 0.866i)10-s + (0.508 − 0.881i)11-s + (0.474 + 1.66i)12-s + (−0.482 − 0.278i)13-s + (3.03 − 1.75i)14-s + (0.422 − 1.67i)15-s + 16-s + (0.748 + 1.29i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.273 − 0.961i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.680 + 0.193i)6-s + (0.662 + 1.14i)7-s + 0.353i·8-s + (−0.849 + 0.526i)9-s + (0.158 − 0.273i)10-s + (0.153 − 0.265i)11-s + (0.136 + 0.480i)12-s + (−0.133 − 0.0772i)13-s + (0.811 − 0.468i)14-s + (0.108 − 0.433i)15-s + 0.250·16-s + (0.181 + 0.314i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.699 + 0.714i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41817 - 0.596300i\)
\(L(\frac12)\) \(\approx\) \(1.41817 - 0.596300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.474 + 1.66i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-5.39 + 1.39i)T \)
good7 \( 1 + (-1.75 - 3.03i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.508 + 0.881i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.482 + 0.278i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.748 - 1.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.58 - 4.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.21T + 23T^{2} \)
29 \( 1 - 2.64T + 29T^{2} \)
37 \( 1 + (-2.08 + 1.20i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.86 - 4.54i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.69 - 3.28i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.95iT - 47T^{2} \)
53 \( 1 + (-1.80 + 3.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.22 - 4.74i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 0.662iT - 61T^{2} \)
67 \( 1 + (-0.768 + 1.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.71 + 5.61i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.23 + 3.02i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.184 + 0.106i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.71 + 2.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.779T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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

−10.08968813082460149206039668180, −9.070953524829885191488624427461, −8.328383351864991406688819452255, −7.61146690205508311211800842788, −6.31476175845522311805214272666, −5.68040231810177954566842080919, −4.80520742349628765310236351645, −3.16269747760785092245864349228, −2.25289643681787595143141287557, −1.25389626053282937927238934447, 0.921361444941150109620751038217, 3.02262774256732607452194602851, 4.40100070589620059820519761722, 4.73329376660031400465182164398, 5.72613975998154321707706014172, 6.78900585014045736715121948016, 7.55548298416758392538692956923, 8.588849891949963182436910670281, 9.371351236032851074686265534204, 10.05252447120408037668141692059

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