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Properties

Label 2-5-5.3-c18-0-5
Degree $2$
Conductor $5$
Sign $-0.835 + 0.549i$
Analytic cond. $10.2693$
Root an. cond. $3.20457$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−17.9 + 17.9i)2-s + (−4.35e3 − 4.35e3i)3-s + 2.61e5i·4-s + (8.23e5 − 1.77e6i)5-s + 1.55e5·6-s + (−2.28e7 + 2.28e7i)7-s + (−9.38e6 − 9.38e6i)8-s − 3.49e8i·9-s + (1.69e7 + 4.64e7i)10-s − 3.44e9·11-s + (1.13e9 − 1.13e9i)12-s + (−1.28e10 − 1.28e10i)13-s − 8.19e8i·14-s + (−1.12e10 + 4.12e9i)15-s − 6.82e10·16-s + (7.38e10 − 7.38e10i)17-s + ⋯
L(s)  = 1  + (−0.0350 + 0.0350i)2-s + (−0.221 − 0.221i)3-s + 0.997i·4-s + (0.421 − 0.906i)5-s + 0.0154·6-s + (−0.566 + 0.566i)7-s + (−0.0699 − 0.0699i)8-s − 0.902i·9-s + (0.0169 + 0.0464i)10-s − 1.46·11-s + (0.220 − 0.220i)12-s + (−1.21 − 1.21i)13-s − 0.0396i·14-s + (−0.293 + 0.107i)15-s − 0.992·16-s + (0.622 − 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(10.2693\)
Root analytic conductor: \(3.20457\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :9),\ -0.835 + 0.549i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.126245 - 0.421528i\)
\(L(\frac12)\) \(\approx\) \(0.126245 - 0.421528i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-8.23e5 + 1.77e6i)T \)
good2 \( 1 + (17.9 - 17.9i)T - 2.62e5iT^{2} \)
3 \( 1 + (4.35e3 + 4.35e3i)T + 3.87e8iT^{2} \)
7 \( 1 + (2.28e7 - 2.28e7i)T - 1.62e15iT^{2} \)
11 \( 1 + 3.44e9T + 5.55e18T^{2} \)
13 \( 1 + (1.28e10 + 1.28e10i)T + 1.12e20iT^{2} \)
17 \( 1 + (-7.38e10 + 7.38e10i)T - 1.40e22iT^{2} \)
19 \( 1 - 2.01e11iT - 1.04e23T^{2} \)
23 \( 1 + (-1.37e12 - 1.37e12i)T + 3.24e24iT^{2} \)
29 \( 1 + 8.04e12iT - 2.10e26T^{2} \)
31 \( 1 + 1.71e13T + 6.99e26T^{2} \)
37 \( 1 + (-5.12e13 + 5.12e13i)T - 1.68e28iT^{2} \)
41 \( 1 + 2.53e14T + 1.07e29T^{2} \)
43 \( 1 + (-1.26e14 - 1.26e14i)T + 2.52e29iT^{2} \)
47 \( 1 + (5.46e14 - 5.46e14i)T - 1.25e30iT^{2} \)
53 \( 1 + (-3.15e15 - 3.15e15i)T + 1.08e31iT^{2} \)
59 \( 1 + 8.70e15iT - 7.50e31T^{2} \)
61 \( 1 + 1.20e16T + 1.36e32T^{2} \)
67 \( 1 + (-1.51e16 + 1.51e16i)T - 7.40e32iT^{2} \)
71 \( 1 - 6.47e15T + 2.10e33T^{2} \)
73 \( 1 + (-6.95e16 - 6.95e16i)T + 3.46e33iT^{2} \)
79 \( 1 + 9.20e16iT - 1.43e34T^{2} \)
83 \( 1 + (9.22e16 + 9.22e16i)T + 3.49e34iT^{2} \)
89 \( 1 + 5.82e16iT - 1.22e35T^{2} \)
97 \( 1 + (-2.01e17 + 2.01e17i)T - 5.77e35iT^{2} \)
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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

−18.23206659362481121070814729875, −17.06874712783733825856315569721, −15.61389801099228906428600515419, −12.96450344406058032309778804915, −12.28331640568229241067676808559, −9.537567095014969572416213647317, −7.75641309891454671453056991457, −5.42072194530761160529871955911, −2.89237217784050611803794623987, −0.19709014139290633029209508183, 2.31651593181911805259673370732, 5.13186284254965361235728532122, 6.97947455377995843458100204240, 9.958561111941882701232182872347, 10.80941470696743461097612851021, 13.52022312699908440382771464511, 14.89023101789098636703139066652, 16.60897584104747658031579385541, 18.55570716607080154587760256058, 19.53965549573607476759711189765

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