Properties

Label 2-31824-1.1-c1-0-16
Degree $2$
Conductor $31824$
Sign $1$
Analytic cond. $254.115$
Root an. cond. $15.9410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 4·11-s + 13-s − 17-s + 4·19-s − 25-s + 2·29-s + 8·31-s − 2·37-s − 2·41-s + 4·43-s + 8·47-s − 7·49-s + 10·53-s + 8·55-s + 4·59-s + 14·61-s + 2·65-s + 4·67-s − 14·73-s + 8·79-s − 4·83-s − 2·85-s + 6·89-s + 8·95-s − 6·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.20·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 49-s + 1.37·53-s + 1.07·55-s + 0.520·59-s + 1.79·61-s + 0.248·65-s + 0.488·67-s − 1.63·73-s + 0.900·79-s − 0.439·83-s − 0.216·85-s + 0.635·89-s + 0.820·95-s − 0.609·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31824\)    =    \(2^{4} \cdot 3^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(254.115\)
Root analytic conductor: \(15.9410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 31824,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.700542831\)
\(L(\frac12)\) \(\approx\) \(3.700542831\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.00775514696786, −14.31796887385678, −14.11215667428918, −13.44096808839433, −13.20703747705707, −12.32573073063147, −11.77988909310708, −11.56784385709540, −10.67489202731927, −10.20508497309687, −9.602644429989829, −9.285749222231189, −8.580362947631242, −8.137424458555636, −7.219381211372168, −6.770376828847969, −6.204992163634711, −5.651250246475473, −5.084028941580542, −4.234052196545388, −3.760136957911145, −2.881002245172081, −2.228547499450869, −1.400180694780998, −0.7962902065217460, 0.7962902065217460, 1.400180694780998, 2.228547499450869, 2.881002245172081, 3.760136957911145, 4.234052196545388, 5.084028941580542, 5.651250246475473, 6.204992163634711, 6.770376828847969, 7.219381211372168, 8.137424458555636, 8.580362947631242, 9.285749222231189, 9.602644429989829, 10.20508497309687, 10.67489202731927, 11.56784385709540, 11.77988909310708, 12.32573073063147, 13.20703747705707, 13.44096808839433, 14.11215667428918, 14.31796887385678, 15.00775514696786

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