↑

Properties

Label 2-930-155.64-c1-0-29
Degree $2$
Conductor $930$
Sign $-0.996 - 0.0886i$
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  + (0.587 − 0.809i)2-s + (−0.587 − 0.809i)3-s + (−0.309 − 0.951i)4-s + (−1.61 − 1.54i)5-s − 6-s + (3.49 − 1.13i)7-s + (−0.951 − 0.309i)8-s + (−0.309 + 0.951i)9-s + (−2.19 + 0.401i)10-s + (−0.348 − 1.07i)11-s + (−0.587 + 0.809i)12-s + (−3.06 − 4.21i)13-s + (1.13 − 3.49i)14-s + (−0.297 + 2.21i)15-s + (−0.809 + 0.587i)16-s + (3.06 + 0.997i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (−0.339 − 0.467i)3-s + (−0.154 − 0.475i)4-s + (−0.723 − 0.690i)5-s − 0.408·6-s + (1.31 − 0.428i)7-s + (−0.336 − 0.109i)8-s + (−0.103 + 0.317i)9-s + (−0.695 + 0.127i)10-s + (−0.105 − 0.323i)11-s + (−0.169 + 0.233i)12-s + (−0.850 − 1.16i)13-s + (0.303 − 0.933i)14-s + (−0.0768 + 0.572i)15-s + (−0.202 + 0.146i)16-s + (0.744 + 0.241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.996 - 0.0886i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.996 - 0.0886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0574941 + 1.29445i\)
\(L(\frac12)\) \(\approx\) \(0.0574941 + 1.29445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (1.61 + 1.54i)T \)
31 \( 1 + (-5.55 + 0.331i)T \)
good7 \( 1 + (-3.49 + 1.13i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.348 + 1.07i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.06 + 4.21i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.06 - 0.997i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.27 + 0.923i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.25 + 0.408i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.55 + 4.03i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 9.34iT - 37T^{2} \)
41 \( 1 + (8.91 + 6.47i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.705 + 0.970i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (-1.10 - 1.52i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.40 + 0.457i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.57 - 5.50i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 0.883T + 61T^{2} \)
67 \( 1 + 4.69iT - 67T^{2} \)
71 \( 1 + (1.31 - 4.06i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.09 - 0.357i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.295 + 0.908i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.67 + 3.67i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.382 + 1.17i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-9.54 + 3.10i)T + (78.4 - 57.0i)T^{2} \)
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

−9.934056818397235837377644678202, −8.550653514707069985824878142863, −7.971922954709868912044964213800, −7.31947559600282552157946549490, −5.85989829310996160239269442406, −5.04883138792925897996352295514, −4.42462506831293913985448064666, −3.18800609180806203662036925081, −1.72389002430616059920152699868, −0.56163625200005609824776508740, 2.10621669096284737929223998937, 3.50593938321753105714246400373, 4.53309195104904653850662125736, 5.06102290846264489994011430365, 6.17741048932394518329061536691, 7.18699031825762012698682730039, 7.77159511323626944012942501979, 8.666167390155414860939364626421, 9.645673044073777417362292717137, 10.62356301244810597338815673938

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