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Properties

Label 2-31-31.5-c1-0-0
Degree $2$
Conductor $31$
Sign $0.654 + 0.755i$
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  − 2.41·2-s + (1.20 − 2.09i)3-s + 3.82·4-s + (−0.5 − 0.866i)5-s + (−2.91 + 5.04i)6-s + (−1.20 + 2.09i)7-s − 4.41·8-s + (−1.41 − 2.44i)9-s + (1.20 + 2.09i)10-s + (2.62 + 4.54i)11-s + (4.62 − 8.00i)12-s + (−0.914 − 1.58i)13-s + (2.91 − 5.04i)14-s − 2.41·15-s + 2.99·16-s + (0.0857 − 0.148i)17-s + ⋯
L(s)  = 1  − 1.70·2-s + (0.696 − 1.20i)3-s + 1.91·4-s + (−0.223 − 0.387i)5-s + (−1.18 + 2.06i)6-s + (−0.456 + 0.790i)7-s − 1.56·8-s + (−0.471 − 0.816i)9-s + (0.381 + 0.661i)10-s + (0.790 + 1.36i)11-s + (1.33 − 2.31i)12-s + (−0.253 − 0.439i)13-s + (0.778 − 1.34i)14-s − 0.623·15-s + 0.749·16-s + (0.0208 − 0.0360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(0.247536\)
Root analytic conductor: \(0.497530\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381984 - 0.174414i\)
\(L(\frac12)\) \(\approx\) \(0.381984 - 0.174414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (5 + 2.44i)T \)
good2 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.20 - 2.09i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.62 - 4.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.914 + 1.58i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.0857 + 0.148i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.792 - 1.37i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.74 + 8.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.44 - 7.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 + (0.0857 + 0.148i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.03 + 8.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 + (-2.62 - 4.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.03 - 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.91 + 3.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.62 + 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.03 + 3.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−17.26816825498542119113283278933, −15.94571492275951373744581470594, −14.65539479419801224756875989542, −12.75642799665712825399666513073, −11.93735652217521342755533904634, −9.928252672436318742225145012911, −8.839704871685090155405517474736, −7.81249002575234326240777543890, −6.69249135921681364561766067139, −2.04312140270729444007225676723, 3.56314932660478550441116226111, 6.86466203132722501192569275906, 8.464004914666357168084196747531, 9.407388892625902277829897186403, 10.40389187153593045943570773306, 11.35507817901893618363075489079, 13.93141751074214876309984148113, 15.19085808359514137694766085875, 16.41724931310326394317818696154, 16.79001358412132854126782658011

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