Properties

Degree 2
Conductor 31
Sign $0.696 + 0.717i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.380 − 1.17i)2-s + (−2.02 − 0.431i)3-s + (0.389 + 0.283i)4-s + (0.772 + 1.33i)5-s + (−1.27 + 2.21i)6-s + (−3.47 + 1.54i)7-s + (2.47 − 1.79i)8-s + (1.19 + 0.531i)9-s + (1.86 − 0.395i)10-s + (0.393 − 3.74i)11-s + (−0.668 − 0.742i)12-s + (−1.76 + 1.95i)13-s + (0.489 + 4.66i)14-s + (−0.991 − 3.05i)15-s + (−0.866 − 2.66i)16-s + (−0.394 − 3.75i)17-s + ⋯
L(s)  = 1  + (0.269 − 0.828i)2-s + (−1.17 − 0.249i)3-s + (0.194 + 0.141i)4-s + (0.345 + 0.598i)5-s + (−0.521 + 0.903i)6-s + (−1.31 + 0.584i)7-s + (0.874 − 0.635i)8-s + (0.397 + 0.177i)9-s + (0.589 − 0.125i)10-s + (0.118 − 1.12i)11-s + (−0.193 − 0.214i)12-s + (−0.489 + 0.543i)13-s + (0.130 + 1.24i)14-s + (−0.255 − 0.787i)15-s + (−0.216 − 0.666i)16-s + (−0.0956 − 0.910i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.696 + 0.717i$
motivic weight  =  \(1\)
character  :  $\chi_{31} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 31,\ (\ :1/2),\ 0.696 + 0.717i)$
$L(1)$  $\approx$  $0.591864 - 0.250216i$
$L(\frac12)$  $\approx$  $0.591864 - 0.250216i$
$L(\frac{3}{2})$   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 \( 1 + (3.88 + 3.98i)T \)
good2 \( 1 + (-0.380 + 1.17i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (2.02 + 0.431i)T + (2.74 + 1.22i)T^{2} \)
5 \( 1 + (-0.772 - 1.33i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.47 - 1.54i)T + (4.68 - 5.20i)T^{2} \)
11 \( 1 + (-0.393 + 3.74i)T + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (1.76 - 1.95i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (0.394 + 3.75i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (-4.08 - 4.53i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (0.736 - 0.534i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.10 - 6.47i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (-0.907 + 1.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.329 - 0.0700i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-2.59 - 2.88i)T + (-4.49 + 42.7i)T^{2} \)
47 \( 1 + (0.367 + 1.13i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.14 + 0.953i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (7.60 + 1.61i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + 2.72T + 61T^{2} \)
67 \( 1 + (-3.71 - 6.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.65 + 2.07i)T + (47.5 + 52.7i)T^{2} \)
73 \( 1 + (0.563 - 5.36i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (-1.01 - 9.68i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-8.21 + 1.74i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (4.12 + 2.99i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (8.82 + 6.41i)T + (29.9 + 92.2i)T^{2} \)
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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

−16.56052478539370329345981975182, −16.19607750501165294556985677550, −14.06624895282221810356105092186, −12.70401889904057804493190581589, −11.84339625733855223979571121151, −10.91609395010105583390893302033, −9.600657510813213688221874409660, −6.94890037032596182305044928658, −5.82483504900735932203209099317, −3.10508788532872281543473427084, 4.87636712298002155565865665085, 6.08672357786122193649285099056, 7.25164698547656826192155015842, 9.674780228125041538017402437501, 10.76293350628672922727970560184, 12.35524577570050580398960861080, 13.46280752281583176758270480092, 15.14606788968162121076561563565, 16.10817007265335702924587462823, 16.97513268454174128175482095678

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