Properties

Degree 2
Conductor 31
Sign $0.532 + 0.846i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.02 − 0.744i)2-s + (−0.155 − 1.47i)3-s + (−0.122 − 0.376i)4-s + (1.90 + 3.29i)5-s + (−0.940 + 1.62i)6-s + (−2.14 − 0.455i)7-s + (−0.937 + 2.88i)8-s + (0.779 − 0.165i)9-s + (0.503 − 4.78i)10-s + (−0.636 + 0.706i)11-s + (−0.536 + 0.238i)12-s + (0.153 + 0.0683i)13-s + (1.85 + 2.06i)14-s + (4.56 − 3.31i)15-s + (2.46 − 1.79i)16-s + (−4.40 − 4.88i)17-s + ⋯
L(s)  = 1  + (−0.724 − 0.526i)2-s + (−0.0895 − 0.852i)3-s + (−0.0611 − 0.188i)4-s + (0.849 + 1.47i)5-s + (−0.383 + 0.664i)6-s + (−0.809 − 0.172i)7-s + (−0.331 + 1.02i)8-s + (0.259 − 0.0552i)9-s + (0.159 − 1.51i)10-s + (−0.191 + 0.212i)11-s + (−0.154 + 0.0689i)12-s + (0.0426 + 0.0189i)13-s + (0.495 + 0.550i)14-s + (1.17 − 0.856i)15-s + (0.617 − 0.448i)16-s + (−1.06 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.532 + 0.846i$
motivic weight  =  \(1\)
character  :  $\chi_{31} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 31,\ (\ :1/2),\ 0.532 + 0.846i)$
$L(1)$  $\approx$  $0.457812 - 0.252720i$
$L(\frac12)$  $\approx$  $0.457812 - 0.252720i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 31$, \(F_p\) is a polynomial of degree 2. If $p = 31$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad31 \( 1 + (-5.56 + 0.217i)T \)
good2 \( 1 + (1.02 + 0.744i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.155 + 1.47i)T + (-2.93 + 0.623i)T^{2} \)
5 \( 1 + (-1.90 - 3.29i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.14 + 0.455i)T + (6.39 + 2.84i)T^{2} \)
11 \( 1 + (0.636 - 0.706i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.153 - 0.0683i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (4.40 + 4.88i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (1.05 - 0.468i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (1.43 - 4.40i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.08 + 0.785i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (-1.93 + 3.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0343 - 0.326i)T + (-40.1 - 8.52i)T^{2} \)
43 \( 1 + (-8.79 + 3.91i)T + (28.7 - 31.9i)T^{2} \)
47 \( 1 + (4.56 - 3.31i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-7.17 + 1.52i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (-0.277 - 2.63i)T + (-57.7 + 12.2i)T^{2} \)
61 \( 1 + 1.74T + 61T^{2} \)
67 \( 1 + (0.276 + 0.478i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.11 - 0.236i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (5.30 - 5.88i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (-3.03 - 3.37i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (0.0341 - 0.324i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + (4.54 + 13.9i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.79 - 14.7i)T + (-78.4 + 57.0i)T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.57230519685699834571213762740, −15.54186649696213514194635211638, −14.12163912903776355253085629117, −13.24951056624638261210816560080, −11.50936399296457131758633707604, −10.29949869700133525473133900653, −9.465046162711892414273042564720, −7.25033543595896290765308511118, −6.15848430541584714162617242182, −2.40987483438552121348599809955, 4.39803154546424988575029186627, 6.24624228753942291185949898939, 8.445352971424050755333431010657, 9.298715774290058246659572030610, 10.25452158532651833472886768391, 12.61735037549026655571640127708, 13.23262330389866719257466806735, 15.43416152318620214421274727383, 16.29065643750444493478152889627, 16.88899268542569971525410508319

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