Properties

Label 2-5-1.1-c7-0-2
Degree $2$
Conductor $5$
Sign $-1$
Analytic cond. $1.56192$
Root an. cond. $1.24977$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14·2-s − 48·3-s + 68·4-s + 125·5-s + 672·6-s − 1.64e3·7-s + 840·8-s + 117·9-s − 1.75e3·10-s + 172·11-s − 3.26e3·12-s + 3.86e3·13-s + 2.30e4·14-s − 6.00e3·15-s − 2.04e4·16-s − 1.22e4·17-s − 1.63e3·18-s − 2.59e4·19-s + 8.50e3·20-s + 7.89e4·21-s − 2.40e3·22-s + 1.29e4·23-s − 4.03e4·24-s + 1.56e4·25-s − 5.40e4·26-s + 9.93e4·27-s − 1.11e5·28-s + ⋯
L(s)  = 1  − 1.23·2-s − 1.02·3-s + 0.531·4-s + 0.447·5-s + 1.27·6-s − 1.81·7-s + 0.580·8-s + 0.0534·9-s − 0.553·10-s + 0.0389·11-s − 0.545·12-s + 0.487·13-s + 2.24·14-s − 0.459·15-s − 1.24·16-s − 0.604·17-s − 0.0662·18-s − 0.867·19-s + 0.237·20-s + 1.85·21-s − 0.0482·22-s + 0.222·23-s − 0.595·24-s + 1/5·25-s − 0.603·26-s + 0.971·27-s − 0.962·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-1$
Analytic conductor: \(1.56192\)
Root analytic conductor: \(1.24977\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - p^{3} T \)
good2 \( 1 + 7 p T + p^{7} T^{2} \)
3 \( 1 + 16 p T + p^{7} T^{2} \)
7 \( 1 + 1644 T + p^{7} T^{2} \)
11 \( 1 - 172 T + p^{7} T^{2} \)
13 \( 1 - 3862 T + p^{7} T^{2} \)
17 \( 1 + 12254 T + p^{7} T^{2} \)
19 \( 1 + 25940 T + p^{7} T^{2} \)
23 \( 1 - 564 p T + p^{7} T^{2} \)
29 \( 1 + 81610 T + p^{7} T^{2} \)
31 \( 1 + 156888 T + p^{7} T^{2} \)
37 \( 1 - 110126 T + p^{7} T^{2} \)
41 \( 1 - 467882 T + p^{7} T^{2} \)
43 \( 1 + 499208 T + p^{7} T^{2} \)
47 \( 1 + 396884 T + p^{7} T^{2} \)
53 \( 1 + 1280498 T + p^{7} T^{2} \)
59 \( 1 + 1337420 T + p^{7} T^{2} \)
61 \( 1 + 923978 T + p^{7} T^{2} \)
67 \( 1 + 797304 T + p^{7} T^{2} \)
71 \( 1 - 5103392 T + p^{7} T^{2} \)
73 \( 1 + 4267478 T + p^{7} T^{2} \)
79 \( 1 + 960 T + p^{7} T^{2} \)
83 \( 1 - 6140832 T + p^{7} T^{2} \)
89 \( 1 - 2010570 T + p^{7} T^{2} \)
97 \( 1 + 4881934 T + p^{7} T^{2} \)
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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

−22.16548253268393328947846819386, −19.79127854535097455062440363103, −18.48308219402311061229510076046, −17.12302975315060672117957157880, −16.19722697511786235709360116310, −13.03645802025512111543084479615, −10.75638735439445487761160215799, −9.274009186839343165913297371932, −6.41815377421923413546724317339, 0, 6.41815377421923413546724317339, 9.274009186839343165913297371932, 10.75638735439445487761160215799, 13.03645802025512111543084479615, 16.19722697511786235709360116310, 17.12302975315060672117957157880, 18.48308219402311061229510076046, 19.79127854535097455062440363103, 22.16548253268393328947846819386

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