↑

Properties

Label 2-31-31.15-c2-0-3
Degree $2$
Conductor $31$
Sign $0.865 + 0.500i$
Analytic cond. $0.844688$
Root an. cond. $0.919069$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 − 0.862i)2-s + (1.08 − 1.49i)3-s + (−0.570 + 1.75i)4-s − 4.14·5-s − 2.70i·6-s + (1.06 − 3.28i)7-s + (2.65 + 8.15i)8-s + (1.72 + 5.32i)9-s + (−4.92 + 3.57i)10-s + (−13.6 − 4.42i)11-s + (2.00 + 2.75i)12-s + (5.41 − 7.45i)13-s + (−1.56 − 4.81i)14-s + (−4.49 + 6.18i)15-s + (4.20 + 3.05i)16-s + (−1.46 + 0.474i)17-s + ⋯
L(s)  = 1  + (0.593 − 0.431i)2-s + (0.361 − 0.497i)3-s + (−0.142 + 0.439i)4-s − 0.829·5-s − 0.451i·6-s + (0.152 − 0.469i)7-s + (0.331 + 1.01i)8-s + (0.192 + 0.591i)9-s + (−0.492 + 0.357i)10-s + (−1.23 − 0.402i)11-s + (0.166 + 0.229i)12-s + (0.416 − 0.573i)13-s + (−0.111 − 0.344i)14-s + (−0.299 + 0.412i)15-s + (0.262 + 0.190i)16-s + (−0.0859 + 0.0279i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(0.844688\)
Root analytic conductor: \(0.919069\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :1),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19784 - 0.321668i\)
\(L(\frac12)\) \(\approx\) \(1.19784 - 0.321668i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (-20.1 - 23.5i)T \)
good2 \( 1 + (-1.18 + 0.862i)T + (1.23 - 3.80i)T^{2} \)
3 \( 1 + (-1.08 + 1.49i)T + (-2.78 - 8.55i)T^{2} \)
5 \( 1 + 4.14T + 25T^{2} \)
7 \( 1 + (-1.06 + 3.28i)T + (-39.6 - 28.8i)T^{2} \)
11 \( 1 + (13.6 + 4.42i)T + (97.8 + 71.1i)T^{2} \)
13 \( 1 + (-5.41 + 7.45i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (1.46 - 0.474i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-16.5 + 12.0i)T + (111. - 343. i)T^{2} \)
23 \( 1 + (-17.2 + 5.61i)T + (427. - 310. i)T^{2} \)
29 \( 1 + (23.9 + 32.9i)T + (-259. + 799. i)T^{2} \)
37 \( 1 - 42.0iT - 1.36e3T^{2} \)
41 \( 1 + (61.3 - 44.6i)T + (519. - 1.59e3i)T^{2} \)
43 \( 1 + (-15.1 - 20.9i)T + (-571. + 1.75e3i)T^{2} \)
47 \( 1 + (3.99 + 2.90i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-38.5 + 12.5i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-36.1 - 26.2i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + 84.8iT - 3.72e3T^{2} \)
67 \( 1 + 109.T + 4.48e3T^{2} \)
71 \( 1 + (19.6 + 60.3i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (85.9 + 27.9i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-142. + 46.2i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-25.0 - 34.4i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + (-46.3 - 15.0i)T + (6.40e3 + 4.65e3i)T^{2} \)
97 \( 1 + (-15.1 + 46.6i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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

−16.50179256336756154780147634031, −15.32148862839194119655827328774, −13.55725102926617197329912957822, −13.24973870191795292769096118279, −11.77501744854709157795768998894, −10.67017008345374636840865929526, −8.253001055265714988724311366538, −7.57271174696019399957114396531, −4.86070483010359271522885110267, −3.06032179972579901000676426090, 3.87015210008785889206195543415, 5.40581355199600429011771461799, 7.28766640311883197655305894584, 9.029025776102650492893136924223, 10.36470965025232442497157425731, 12.00175416313450754070398032673, 13.37246812754058780780773902890, 14.72932914697800156300031917484, 15.44785654529404936109673830088, 16.12697560008465348083490418722

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