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Properties

Label 2-5-5.2-c12-0-0
Degree $2$
Conductor $5$
Sign $0.867 - 0.498i$
Analytic cond. $4.56996$
Root an. cond. $2.13774$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−54.8 − 54.8i)2-s + (−506. + 506. i)3-s + 1.91e3i·4-s + (1.56e4 + 502. i)5-s + 5.55e4·6-s + (7.85e4 + 7.85e4i)7-s + (−1.19e5 + 1.19e5i)8-s + 1.85e4i·9-s + (−8.28e5 − 8.84e5i)10-s + 1.27e6·11-s + (−9.72e5 − 9.72e5i)12-s + (−3.53e6 + 3.53e6i)13-s − 8.61e6i·14-s + (−8.16e6 + 7.65e6i)15-s + 2.09e7·16-s + (2.56e7 + 2.56e7i)17-s + ⋯
L(s)  = 1  + (−0.856 − 0.856i)2-s + (−0.694 + 0.694i)3-s + 0.468i·4-s + (0.999 + 0.0321i)5-s + 1.19·6-s + (0.667 + 0.667i)7-s + (−0.455 + 0.455i)8-s + 0.0348i·9-s + (−0.828 − 0.884i)10-s + 0.717·11-s + (−0.325 − 0.325i)12-s + (−0.732 + 0.732i)13-s − 1.14i·14-s + (−0.716 + 0.671i)15-s + 1.24·16-s + (1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.867 - 0.498i$
Analytic conductor: \(4.56996\)
Root analytic conductor: \(2.13774\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :6),\ 0.867 - 0.498i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.845062 + 0.225439i\)
\(L(\frac12)\) \(\approx\) \(0.845062 + 0.225439i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.56e4 - 502. i)T \)
good2 \( 1 + (54.8 + 54.8i)T + 4.09e3iT^{2} \)
3 \( 1 + (506. - 506. i)T - 5.31e5iT^{2} \)
7 \( 1 + (-7.85e4 - 7.85e4i)T + 1.38e10iT^{2} \)
11 \( 1 - 1.27e6T + 3.13e12T^{2} \)
13 \( 1 + (3.53e6 - 3.53e6i)T - 2.32e13iT^{2} \)
17 \( 1 + (-2.56e7 - 2.56e7i)T + 5.82e14iT^{2} \)
19 \( 1 + 5.31e7iT - 2.21e15T^{2} \)
23 \( 1 + (1.16e8 - 1.16e8i)T - 2.19e16iT^{2} \)
29 \( 1 - 8.95e8iT - 3.53e17T^{2} \)
31 \( 1 + 1.69e8T + 7.87e17T^{2} \)
37 \( 1 + (6.45e8 + 6.45e8i)T + 6.58e18iT^{2} \)
41 \( 1 - 1.87e9T + 2.25e19T^{2} \)
43 \( 1 + (-4.24e9 + 4.24e9i)T - 3.99e19iT^{2} \)
47 \( 1 + (3.40e9 + 3.40e9i)T + 1.16e20iT^{2} \)
53 \( 1 + (-2.01e10 + 2.01e10i)T - 4.91e20iT^{2} \)
59 \( 1 - 4.54e10iT - 1.77e21T^{2} \)
61 \( 1 + 3.24e10T + 2.65e21T^{2} \)
67 \( 1 + (1.79e10 + 1.79e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 9.83e9T + 1.64e22T^{2} \)
73 \( 1 + (1.11e10 - 1.11e10i)T - 2.29e22iT^{2} \)
79 \( 1 + 1.98e11iT - 5.90e22T^{2} \)
83 \( 1 + (-1.02e11 + 1.02e11i)T - 1.06e23iT^{2} \)
89 \( 1 - 1.71e11iT - 2.46e23T^{2} \)
97 \( 1 + (-2.72e11 - 2.72e11i)T + 6.93e23iT^{2} \)
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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

−21.35557263108676058792127550196, −19.49162257571797221985827529623, −17.91051244668513764092662766345, −16.90076428338586109620955559288, −14.58022601634052849182325089562, −11.84039762648628228594715533948, −10.45562221180203111283632896641, −9.145432583560082006761749820381, −5.45458245200783028872203987211, −1.82377008176631433469001604822, 0.885715920460734114579811008553, 6.03371980900105938852949465919, 7.60986471642206963434208537514, 9.824604388936127300177876936312, 12.27122290047480987449832421511, 14.37390737210262697546168549072, 16.76225490213138180945404750794, 17.50423484840567955791873033627, 18.48776719854799839858414151935, 20.83139075018064926804485900889

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