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Properties

Label 2-279-31.4-c1-0-2
Degree $2$
Conductor $279$
Sign $-0.610 + 0.792i$
Analytic cond. $2.22782$
Root an. cond. $1.49259$
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  + (0.833 + 2.56i)2-s + (−4.26 + 3.10i)4-s − 3.42·5-s + (1.97 − 1.43i)7-s + (−7.15 − 5.19i)8-s + (−2.85 − 8.77i)10-s + (−3.56 + 2.58i)11-s + (−1.31 + 4.05i)13-s + (5.32 + 3.87i)14-s + (4.10 − 12.6i)16-s + (3.00 + 2.18i)17-s + (0.445 + 1.37i)19-s + (14.6 − 10.6i)20-s + (−9.60 − 6.97i)22-s + (1.84 + 1.34i)23-s + ⋯
L(s)  = 1  + (0.589 + 1.81i)2-s + (−2.13 + 1.55i)4-s − 1.52·5-s + (0.746 − 0.542i)7-s + (−2.52 − 1.83i)8-s + (−0.901 − 2.77i)10-s + (−1.07 + 0.779i)11-s + (−0.365 + 1.12i)13-s + (1.42 + 1.03i)14-s + (1.02 − 3.15i)16-s + (0.728 + 0.529i)17-s + (0.102 + 0.314i)19-s + (3.26 − 2.37i)20-s + (−2.04 − 1.48i)22-s + (0.385 + 0.280i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $-0.610 + 0.792i$
Analytic conductor: \(2.22782\)
Root analytic conductor: \(1.49259\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{279} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 279,\ (\ :1/2),\ -0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368343 - 0.748619i\)
\(L(\frac12)\) \(\approx\) \(0.368343 - 0.748619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (4.47 - 3.31i)T \)
good2 \( 1 + (-0.833 - 2.56i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + 3.42T + 5T^{2} \)
7 \( 1 + (-1.97 + 1.43i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (3.56 - 2.58i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.31 - 4.05i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.00 - 2.18i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.445 - 1.37i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.84 - 1.34i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.696 - 2.14i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 2.58T + 37T^{2} \)
41 \( 1 + (1.47 + 4.54i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.68 + 5.19i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.0697 - 0.214i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-8.96 - 6.51i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.10 - 3.38i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 9.41T + 61T^{2} \)
67 \( 1 - 8.78T + 67T^{2} \)
71 \( 1 + (1.85 + 1.34i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.92 + 3.58i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-13.6 - 9.90i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.26 + 6.95i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.5 + 8.37i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (12.8 - 9.35i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.55398062878113550989938993092, −11.97040483878691962107135812530, −10.62248607044793962414346440193, −9.091945874158147205845419387244, −8.024746063392519893891630298572, −7.55266286528035478051869878932, −6.91546374698180463282311269113, −5.31008889305265747423312695257, −4.50850864697190089903873287759, −3.68520254786517973119690251500, 0.53862403165090236132012666874, 2.66124880440266245218504153731, 3.50176965074680509832242300931, 4.79651344934394229982298626374, 5.47378300231485144341669659073, 7.82504581456050135615201935893, 8.425198494827642282937205808788, 9.737892438421215801008626088773, 10.84021574249194916912021545446, 11.30331765226842306514718124661

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