Properties

Degree 2
Conductor 31
Sign $0.328 - 0.944i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.571 + 1.75i)2-s + (−0.488 − 0.103i)3-s + (−1.15 − 0.836i)4-s + (−0.603 − 1.04i)5-s + (0.461 − 0.800i)6-s + (3.41 − 1.51i)7-s + (−0.863 + 0.627i)8-s + (−2.51 − 1.11i)9-s + (2.18 − 0.464i)10-s + (0.194 − 1.84i)11-s + (0.475 + 0.528i)12-s + (−3.46 + 3.85i)13-s + (0.721 + 6.86i)14-s + (0.186 + 0.573i)15-s + (−1.48 − 4.58i)16-s + (0.592 + 5.63i)17-s + ⋯
L(s)  = 1  + (−0.404 + 1.24i)2-s + (−0.282 − 0.0599i)3-s + (−0.575 − 0.418i)4-s + (−0.269 − 0.467i)5-s + (0.188 − 0.326i)6-s + (1.28 − 0.573i)7-s + (−0.305 + 0.221i)8-s + (−0.837 − 0.372i)9-s + (0.690 − 0.146i)10-s + (0.0585 − 0.556i)11-s + (0.137 + 0.152i)12-s + (−0.962 + 1.06i)13-s + (0.192 + 1.83i)14-s + (0.0481 + 0.148i)15-s + (−0.372 − 1.14i)16-s + (0.143 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.328 - 0.944i$
motivic weight  =  \(1\)
character  :  $\chi_{31} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 31,\ (\ :1/2),\ 0.328 - 0.944i)$
$L(1)$  $\approx$  $0.461135 + 0.327770i$
$L(\frac12)$  $\approx$  $0.461135 + 0.327770i$
$L(\frac{3}{2})$   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 \( 1 + (1.81 - 5.26i)T \)
good2 \( 1 + (0.571 - 1.75i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.488 + 0.103i)T + (2.74 + 1.22i)T^{2} \)
5 \( 1 + (0.603 + 1.04i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.41 + 1.51i)T + (4.68 - 5.20i)T^{2} \)
11 \( 1 + (-0.194 + 1.84i)T + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (3.46 - 3.85i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (-0.592 - 5.63i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (0.962 + 1.06i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (-2.86 + 2.08i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.424 + 1.30i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (2.25 - 3.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.61 + 0.981i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-4.38 - 4.87i)T + (-4.49 + 42.7i)T^{2} \)
47 \( 1 + (1.30 + 4.02i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (11.8 + 5.27i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (-2.13 - 0.453i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + 2.68T + 61T^{2} \)
67 \( 1 + (1.44 + 2.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.22 - 3.66i)T + (47.5 + 52.7i)T^{2} \)
73 \( 1 + (-0.439 + 4.18i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (1.17 + 11.1i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-10.2 + 2.17i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (-2.18 - 1.58i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.70 - 4.87i)T + (29.9 + 92.2i)T^{2} \)
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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

−17.11681955971301396506816709424, −16.37980472836469985558065010034, −14.78964187204761275895953357358, −14.25543548812310783590304748979, −12.17007710885031703822306354357, −11.03433442204872852542808582688, −8.870270747005983952118131581241, −7.988921498470544467528313885202, −6.50114856772992223337955400878, −4.87106169569145169939996010039, 2.63129835239756389634579536525, 5.22379061394187740150474457877, 7.69038521508176830692022006880, 9.307519786002191128107663902690, 10.80332485836872513170851999697, 11.48166925227993287258754815304, 12.48658249053340011875077368244, 14.41043003583607054977161616318, 15.36587673522949301859522953738, 17.30145543880921557935682019049

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