Properties

Degree 2
Conductor 31
Sign $0.951 - 0.309i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.284 − 0.206i)2-s + (0.302 + 2.87i)3-s + (−0.579 − 1.78i)4-s + (−1.48 − 2.57i)5-s + (0.508 − 0.880i)6-s + (1.05 + 0.224i)7-s + (−0.420 + 1.29i)8-s + (−5.25 + 1.11i)9-s + (−0.109 + 1.03i)10-s + (−1.62 + 1.80i)11-s + (4.95 − 2.20i)12-s + (2.62 + 1.16i)13-s + (−0.254 − 0.282i)14-s + (6.95 − 5.05i)15-s + (−2.64 + 1.92i)16-s + (−1.22 − 1.35i)17-s + ⋯
L(s)  = 1  + (−0.201 − 0.146i)2-s + (0.174 + 1.66i)3-s + (−0.289 − 0.892i)4-s + (−0.664 − 1.15i)5-s + (0.207 − 0.359i)6-s + (0.400 + 0.0850i)7-s + (−0.148 + 0.458i)8-s + (−1.75 + 0.372i)9-s + (−0.0345 + 0.328i)10-s + (−0.490 + 0.544i)11-s + (1.43 − 0.637i)12-s + (0.728 + 0.324i)13-s + (−0.0680 − 0.0755i)14-s + (1.79 − 1.30i)15-s + (−0.662 + 0.481i)16-s + (−0.296 − 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.951 - 0.309i$
motivic weight  =  \(1\)
character  :  $\chi_{31} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 31,\ (\ :1/2),\ 0.951 - 0.309i)$
$L(1)$  $\approx$  $0.608503 + 0.0963875i$
$L(\frac12)$  $\approx$  $0.608503 + 0.0963875i$
$L(\frac{3}{2})$   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 \( 1 + (-1.15 + 5.44i)T \)
good2 \( 1 + (0.284 + 0.206i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.302 - 2.87i)T + (-2.93 + 0.623i)T^{2} \)
5 \( 1 + (1.48 + 2.57i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.05 - 0.224i)T + (6.39 + 2.84i)T^{2} \)
11 \( 1 + (1.62 - 1.80i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-2.62 - 1.16i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (1.22 + 1.35i)T + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (-1.93 + 0.861i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (0.136 - 0.420i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.55 - 1.85i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (1.57 - 2.72i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.726 + 6.90i)T + (-40.1 - 8.52i)T^{2} \)
43 \( 1 + (7.68 - 3.42i)T + (28.7 - 31.9i)T^{2} \)
47 \( 1 + (-6.44 + 4.67i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.86 + 1.03i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (-1.25 - 11.9i)T + (-57.7 + 12.2i)T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + (-3.21 - 5.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.64 - 0.348i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (-9.60 + 10.6i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (2.30 + 2.56i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (1.34 - 12.7i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + (-0.698 - 2.14i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.05 - 3.24i)T + (-78.4 + 57.0i)T^{2} \)
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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

−16.69979795093283094551652174886, −15.70397139970093128788507310776, −15.08208185583504755854330897662, −13.66760659722908943365945743033, −11.69591286859460951933347980614, −10.52050371091663584542996772945, −9.357480875971394550951632159642, −8.484820040391390634739017364592, −5.24863161269803209441645806917, −4.34779856238566896989447204291, 3.14476156855467306061134251707, 6.55704855781521969026441912023, 7.66613091419733596025405588041, 8.384095418537300511020442169262, 11.01622009240806556270779900393, 12.11352281552684703140896079779, 13.26756003355815319076409314953, 14.19668631854590490650345258662, 15.76543468133546268178183389014, 17.38708594726758874646731647909

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