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Properties

Label 2-5-5.3-c26-0-11
Degree $2$
Conductor $5$
Sign $0.207 - 0.978i$
Analytic cond. $21.4146$
Root an. cond. $4.62759$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.03e4 − 1.03e4i)2-s + (−1.27e6 − 1.27e6i)3-s − 1.48e8i·4-s + (4.12e8 + 1.14e9i)5-s − 2.63e10·6-s + (−8.34e10 + 8.34e10i)7-s + (−8.47e11 − 8.47e11i)8-s + 6.84e11i·9-s + (1.62e13 + 7.65e12i)10-s − 5.94e13·11-s + (−1.88e14 + 1.88e14i)12-s + (1.83e14 + 1.83e14i)13-s + 1.73e15i·14-s + (9.35e14 − 1.98e15i)15-s − 7.63e15·16-s + (3.11e15 − 3.11e15i)17-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)2-s + (−0.796 − 0.796i)3-s − 2.21i·4-s + (0.337 + 0.941i)5-s − 2.02·6-s + (−0.861 + 0.861i)7-s + (−1.54 − 1.54i)8-s + 0.269i·9-s + (1.62 + 0.765i)10-s − 1.72·11-s + (−1.76 + 1.76i)12-s + (0.606 + 0.606i)13-s + 2.18i·14-s + (0.480 − 1.01i)15-s − 1.69·16-s + (0.314 − 0.314i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(21.4146\)
Root analytic conductor: \(4.62759\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :13),\ 0.207 - 0.978i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.255053 + 0.206622i\)
\(L(\frac12)\) \(\approx\) \(0.255053 + 0.206622i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.12e8 - 1.14e9i)T \)
good2 \( 1 + (-1.03e4 + 1.03e4i)T - 6.71e7iT^{2} \)
3 \( 1 + (1.27e6 + 1.27e6i)T + 2.54e12iT^{2} \)
7 \( 1 + (8.34e10 - 8.34e10i)T - 9.38e21iT^{2} \)
11 \( 1 + 5.94e13T + 1.19e27T^{2} \)
13 \( 1 + (-1.83e14 - 1.83e14i)T + 9.17e28iT^{2} \)
17 \( 1 + (-3.11e15 + 3.11e15i)T - 9.81e31iT^{2} \)
19 \( 1 + 1.75e16iT - 1.76e33T^{2} \)
23 \( 1 + (3.56e17 + 3.56e17i)T + 2.54e35iT^{2} \)
29 \( 1 + 5.19e18iT - 1.05e38T^{2} \)
31 \( 1 + 5.43e18T + 5.96e38T^{2} \)
37 \( 1 + (2.53e20 - 2.53e20i)T - 5.93e40iT^{2} \)
41 \( 1 + 4.61e20T + 8.55e41T^{2} \)
43 \( 1 + (7.85e20 + 7.85e20i)T + 2.95e42iT^{2} \)
47 \( 1 + (-1.49e21 + 1.49e21i)T - 2.98e43iT^{2} \)
53 \( 1 + (2.72e22 + 2.72e22i)T + 6.77e44iT^{2} \)
59 \( 1 - 9.06e22iT - 1.10e46T^{2} \)
61 \( 1 - 2.72e22T + 2.62e46T^{2} \)
67 \( 1 + (-6.29e23 + 6.29e23i)T - 3.00e47iT^{2} \)
71 \( 1 + 1.59e24T + 1.35e48T^{2} \)
73 \( 1 + (-1.12e24 - 1.12e24i)T + 2.79e48iT^{2} \)
79 \( 1 - 3.79e24iT - 2.17e49T^{2} \)
83 \( 1 + (2.00e24 + 2.00e24i)T + 7.87e49iT^{2} \)
89 \( 1 + 6.85e24iT - 4.83e50T^{2} \)
97 \( 1 + (-6.92e25 + 6.92e25i)T - 4.52e51iT^{2} \)
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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

−15.46974272861775115160773660574, −13.66107193964731957466778400168, −12.62355478349703666329978022582, −11.45733996416216640271166859308, −10.13761102198188677964531240928, −6.48447282032779137017912031752, −5.45499760398897674829600989383, −3.13328587676206936525671537838, −2.04532632956093309269258838045, −0.07734295654355010117846533639, 3.67561438867709905699122107663, 5.05745673015400178870320322545, 5.87202798137126698234152341140, 7.84910167684342017996236868304, 10.32036298354913803664309935235, 12.70318814135049340449132380547, 13.59184535349557736737261864156, 15.81203246445011014069294315901, 16.21636038220481183908340491212, 17.40208759159572260361492230700

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