Properties

Degree 2
Conductor 31
Sign $0.100 + 0.994i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 − 2.55i)2-s + (0.949 + 1.05i)3-s + (−4.23 + 3.07i)4-s + (−0.304 − 0.526i)5-s + (1.90 − 3.30i)6-s + (0.180 + 1.71i)7-s + (7.04 + 5.11i)8-s + (0.103 − 0.980i)9-s + (−1.09 + 1.21i)10-s + (−1.22 − 0.543i)11-s + (−7.26 − 1.54i)12-s + (−3.59 + 0.763i)13-s + (4.24 − 1.88i)14-s + (0.266 − 0.821i)15-s + (4.00 − 12.3i)16-s + (−2.52 + 1.12i)17-s + ⋯
L(s)  = 1  + (−0.587 − 1.80i)2-s + (0.548 + 0.608i)3-s + (−2.11 + 1.53i)4-s + (−0.136 − 0.235i)5-s + (0.779 − 1.34i)6-s + (0.0682 + 0.649i)7-s + (2.49 + 1.80i)8-s + (0.0343 − 0.326i)9-s + (−0.346 + 0.384i)10-s + (−0.368 − 0.164i)11-s + (−2.09 − 0.446i)12-s + (−0.995 + 0.211i)13-s + (1.13 − 0.504i)14-s + (0.0688 − 0.212i)15-s + (1.00 − 3.07i)16-s + (−0.612 + 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.100 + 0.994i$
motivic weight  =  \(1\)
character  :  $\chi_{31} (20, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 31,\ (\ :1/2),\ 0.100 + 0.994i)$
$L(1)$  $\approx$  $0.421231 - 0.380912i$
$L(\frac12)$  $\approx$  $0.421231 - 0.380912i$
$L(\frac{3}{2})$   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 \( 1 + (-4.75 - 2.90i)T \)
good2 \( 1 + (0.831 + 2.55i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.949 - 1.05i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + (0.304 + 0.526i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.180 - 1.71i)T + (-6.84 + 1.45i)T^{2} \)
11 \( 1 + (1.22 + 0.543i)T + (7.36 + 8.17i)T^{2} \)
13 \( 1 + (3.59 - 0.763i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (2.52 - 1.12i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (-2.51 - 0.533i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-0.436 - 0.316i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.51 + 7.73i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (3.87 - 6.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0696 - 0.0773i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (-2.93 - 0.624i)T + (39.2 + 17.4i)T^{2} \)
47 \( 1 + (2.07 - 6.39i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.292 - 2.78i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (-0.311 - 0.346i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 + (4.14 + 7.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.497 - 4.73i)T + (-69.4 - 14.7i)T^{2} \)
73 \( 1 + (-6.85 - 3.05i)T + (48.8 + 54.2i)T^{2} \)
79 \( 1 + (8.86 - 3.94i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (11.1 - 12.3i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-12.3 + 9.00i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.03 + 0.751i)T + (29.9 - 92.2i)T^{2} \)
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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

−17.15993237480583111895567658890, −15.48030569263229066459501564901, −13.96391855273601576140838083312, −12.53898493521394067076294320627, −11.68620157430129660091283636139, −10.20780923203791310064248812100, −9.302103639724299121075745303505, −8.280258001424717085719593983711, −4.47635727147183126633039920895, −2.76877362851943597799652067756, 5.05556253852467870297968779161, 7.07778108851165145153692982402, 7.59217921898349674971064739441, 8.981180771843397048315828955965, 10.44539546789980729814657695441, 13.08166356276084519983698211672, 14.03952722699053288994477472144, 14.94274206043601689669106904098, 16.13030393842019918398268127262, 17.17295768926729791352074520125

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