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Properties

Label 2-31-31.2-c1-0-0
Degree $2$
Conductor $31$
Sign $0.569 + 0.822i$
Analytic cond. $0.247536$
Root an. cond. $0.497530$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 0.951i)2-s + (0.809 − 0.587i)3-s + (0.190 + 0.587i)4-s − 0.381·5-s − 1.61·6-s + (0.927 + 2.85i)7-s + (−0.690 + 2.12i)8-s + (−0.618 + 1.90i)9-s + (0.5 + 0.363i)10-s + (−1.61 − 4.97i)11-s + (0.5 + 0.363i)12-s + (1.5 − 1.08i)13-s + (1.5 − 4.61i)14-s + (−0.309 + 0.224i)15-s + (3.92 − 2.85i)16-s + (−1.30 + 4.02i)17-s + ⋯
L(s)  = 1  + (−0.925 − 0.672i)2-s + (0.467 − 0.339i)3-s + (0.0954 + 0.293i)4-s − 0.170·5-s − 0.660·6-s + (0.350 + 1.07i)7-s + (−0.244 + 0.751i)8-s + (−0.206 + 0.634i)9-s + (0.158 + 0.114i)10-s + (−0.487 − 1.50i)11-s + (0.144 + 0.104i)12-s + (0.416 − 0.302i)13-s + (0.400 − 1.23i)14-s + (−0.0797 + 0.0579i)15-s + (0.981 − 0.713i)16-s + (−0.317 + 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(0.247536\)
Root analytic conductor: \(0.497530\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :1/2),\ 0.569 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459340 - 0.240714i\)
\(L(\frac12)\) \(\approx\) \(0.459340 - 0.240714i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (-0.0450 + 5.56i)T \)
good2 \( 1 + (1.30 + 0.951i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
7 \( 1 + (-0.927 - 2.85i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.61 + 4.97i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.5 + 1.08i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.30 - 4.02i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.04 + 2.93i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.07 + 3.30i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.16 - 3.75i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + (-2 - 1.45i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.92 - 1.40i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.54 + 3.30i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.218 + 0.673i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.427 - 0.310i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 0.236T + 67T^{2} \)
71 \( 1 + (3.42 - 10.5i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.57 - 10.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.73 + 4.16i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.66 + 8.19i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.78 + 17.7i)T + (-78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.15558693051889425009584961495, −15.65227990105115757821855653139, −14.35983335753380228585460522268, −13.03269120146720320449266772396, −11.42547298108203992927067657943, −10.58749408409296169775567131946, −8.593274253077195400660673104268, −8.391174888530887729454457221271, −5.69417246556555821462298156022, −2.48296375840074006334440528554, 4.10831542512091253099542963216, 6.86513297011710528604701685740, 7.949373621221349143315831768295, 9.305256134931976375631630499151, 10.34083536217138989203461251263, 12.22917086864965877804078583878, 13.81964452554547720117898616488, 15.15335401174753181297096234603, 15.99088356063220097422579619443, 17.41029481609552808505357583127

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