Properties

Label 2-5-5.3-c26-0-2
Degree $2$
Conductor $5$
Sign $-0.280 - 0.959i$
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  + (−2.62e3 + 2.62e3i)2-s + (−8.23e5 − 8.23e5i)3-s + 5.33e7i·4-s + (9.07e8 + 8.16e8i)5-s + 4.31e9·6-s + (3.09e10 − 3.09e10i)7-s + (−3.15e11 − 3.15e11i)8-s − 1.18e12i·9-s + (−4.51e12 + 2.36e11i)10-s + 1.89e13·11-s + (4.39e13 − 4.39e13i)12-s + (1.89e14 + 1.89e14i)13-s + 1.62e14i·14-s + (−7.41e13 − 1.41e15i)15-s − 1.92e15·16-s + (−2.10e14 + 2.10e14i)17-s + ⋯
L(s)  = 1  + (−0.319 + 0.319i)2-s + (−0.516 − 0.516i)3-s + 0.795i·4-s + (0.743 + 0.669i)5-s + 0.330·6-s + (0.319 − 0.319i)7-s + (−0.574 − 0.574i)8-s − 0.466i·9-s + (−0.451 + 0.0236i)10-s + 0.547·11-s + (0.410 − 0.410i)12-s + (0.625 + 0.625i)13-s + 0.204i·14-s + (−0.0380 − 0.729i)15-s − 0.427·16-s + (−0.0212 + 0.0212i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.280 - 0.959i$
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: \((2,\ 5,\ (\ :13),\ -0.280 - 0.959i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.800703 + 1.06785i\)
\(L(\frac12)\) \(\approx\) \(0.800703 + 1.06785i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-9.07e8 - 8.16e8i)T \)
good2 \( 1 + (2.62e3 - 2.62e3i)T - 6.71e7iT^{2} \)
3 \( 1 + (8.23e5 + 8.23e5i)T + 2.54e12iT^{2} \)
7 \( 1 + (-3.09e10 + 3.09e10i)T - 9.38e21iT^{2} \)
11 \( 1 - 1.89e13T + 1.19e27T^{2} \)
13 \( 1 + (-1.89e14 - 1.89e14i)T + 9.17e28iT^{2} \)
17 \( 1 + (2.10e14 - 2.10e14i)T - 9.81e31iT^{2} \)
19 \( 1 - 4.30e16iT - 1.76e33T^{2} \)
23 \( 1 + (-1.72e17 - 1.72e17i)T + 2.54e35iT^{2} \)
29 \( 1 - 1.50e19iT - 1.05e38T^{2} \)
31 \( 1 + 3.13e19T + 5.96e38T^{2} \)
37 \( 1 + (-1.99e20 + 1.99e20i)T - 5.93e40iT^{2} \)
41 \( 1 + 1.03e21T + 8.55e41T^{2} \)
43 \( 1 + (-1.56e21 - 1.56e21i)T + 2.95e42iT^{2} \)
47 \( 1 + (3.40e21 - 3.40e21i)T - 2.98e43iT^{2} \)
53 \( 1 + (-8.33e21 - 8.33e21i)T + 6.77e44iT^{2} \)
59 \( 1 + 1.44e23iT - 1.10e46T^{2} \)
61 \( 1 - 1.05e22T + 2.62e46T^{2} \)
67 \( 1 + (6.23e23 - 6.23e23i)T - 3.00e47iT^{2} \)
71 \( 1 - 4.45e22T + 1.35e48T^{2} \)
73 \( 1 + (-1.13e24 - 1.13e24i)T + 2.79e48iT^{2} \)
79 \( 1 + 4.71e24iT - 2.17e49T^{2} \)
83 \( 1 + (9.83e23 + 9.83e23i)T + 7.87e49iT^{2} \)
89 \( 1 - 3.21e25iT - 4.83e50T^{2} \)
97 \( 1 + (1.34e25 - 1.34e25i)T - 4.52e51iT^{2} \)
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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

−17.68041858437844200011328430196, −16.55676963108657393964197124331, −14.43647742438524062603587288037, −12.76402514559392262815389134366, −11.25518637528525892015001889108, −9.194542723895962941621578477662, −7.26035930307137519187717336058, −6.16784127618636825609186383617, −3.55865880589554884202435047677, −1.44802303735002378287241510434, 0.59502034631779663090965763805, 2.04157317562075568728123148283, 4.85416216064197259132595482951, 5.92204718753345073560870516064, 8.814310071265408618096089393818, 10.14823947887304388625329814270, 11.41738887236038573553263892350, 13.52112438040437311063021652009, 15.27812610541089774334963767336, 16.88645767534579453295832878588

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