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Properties

Label 2-13860-1.1-c1-0-24
Degree $2$
Conductor $13860$
Sign $-1$
Analytic cond. $110.672$
Root an. cond. $10.5201$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 11-s + 2·13-s + 3·17-s − 7·19-s + 3·23-s + 25-s + 3·29-s − 4·31-s + 35-s − 10·37-s − 7·43-s + 49-s − 9·53-s − 55-s − 3·59-s − 61-s + 2·65-s + 2·67-s + 12·71-s − 4·73-s − 77-s + 2·79-s + 3·83-s + 3·85-s − 9·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s + 0.727·17-s − 1.60·19-s + 0.625·23-s + 1/5·25-s + 0.557·29-s − 0.718·31-s + 0.169·35-s − 1.64·37-s − 1.06·43-s + 1/7·49-s − 1.23·53-s − 0.134·55-s − 0.390·59-s − 0.128·61-s + 0.248·65-s + 0.244·67-s + 1.42·71-s − 0.468·73-s − 0.113·77-s + 0.225·79-s + 0.329·83-s + 0.325·85-s − 0.953·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13860\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(110.672\)
Root analytic conductor: \(10.5201\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−16.52517808513535, −15.87679202170458, −15.24884472118410, −14.80220552554869, −14.18686428647000, −13.67939358167377, −13.09151361862995, −12.49435451847805, −12.08702111555441, −11.09798481598495, −10.83320134245478, −10.24362231288997, −9.572177079177863, −8.880593307765940, −8.335786516204424, −7.872741117850684, −6.850670770910074, −6.565925964955331, −5.642128682546584, −5.181709837583153, −4.426581273200705, −3.628180658003267, −2.887657655718719, −1.961704674177268, −1.318350378837528, 0, 1.318350378837528, 1.961704674177268, 2.887657655718719, 3.628180658003267, 4.426581273200705, 5.181709837583153, 5.642128682546584, 6.565925964955331, 6.850670770910074, 7.872741117850684, 8.335786516204424, 8.880593307765940, 9.572177079177863, 10.24362231288997, 10.83320134245478, 11.09798481598495, 12.08702111555441, 12.49435451847805, 13.09151361862995, 13.67939358167377, 14.18686428647000, 14.80220552554869, 15.24884472118410, 15.87679202170458, 16.52517808513535

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