Properties

Label 2-134640-1.1-c1-0-119
Degree $2$
Conductor $134640$
Sign $-1$
Analytic cond. $1075.10$
Root an. cond. $32.7888$
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 + 2·7-s + 11-s − 17-s − 4·23-s + 25-s − 2·29-s − 4·31-s + 2·35-s − 2·37-s + 6·43-s − 3·49-s + 6·53-s + 55-s + 14·59-s − 2·61-s − 14·67-s − 2·71-s + 16·73-s + 2·77-s − 12·79-s − 85-s + 18·89-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.301·11-s − 0.242·17-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s + 0.914·43-s − 3/7·49-s + 0.824·53-s + 0.134·55-s + 1.82·59-s − 0.256·61-s − 1.71·67-s − 0.237·71-s + 1.87·73-s + 0.227·77-s − 1.35·79-s − 0.108·85-s + 1.90·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(134640\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(1075.10\)
Root analytic conductor: \(32.7888\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{134640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 134640,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−13.67513426901731, −13.29899814589354, −12.71329991725044, −12.27766499215631, −11.65738618597528, −11.39060343703134, −10.77664023758001, −10.34078327217848, −9.875153961121361, −9.235852635805091, −8.895817481026905, −8.341452773823202, −7.826333416997248, −7.282930198647393, −6.858589451914842, −6.020584557295463, −5.889896533449271, −5.047496295897169, −4.784431216918810, −3.893308312981187, −3.699129173602159, −2.693916714896527, −2.176187743157044, −1.632671986224957, −0.9591276634314386, 0, 0.9591276634314386, 1.632671986224957, 2.176187743157044, 2.693916714896527, 3.699129173602159, 3.893308312981187, 4.784431216918810, 5.047496295897169, 5.889896533449271, 6.020584557295463, 6.858589451914842, 7.282930198647393, 7.826333416997248, 8.341452773823202, 8.895817481026905, 9.235852635805091, 9.875153961121361, 10.34078327217848, 10.77664023758001, 11.39060343703134, 11.65738618597528, 12.27766499215631, 12.71329991725044, 13.29899814589354, 13.67513426901731

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