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  :  $\chi_{31} (14, \cdot )$
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

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

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