Properties

Label 2-134640-1.1-c1-0-80
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 − 11-s + 6·13-s − 17-s − 4·19-s + 25-s − 6·29-s − 2·37-s + 6·41-s + 4·43-s − 8·47-s − 7·49-s − 14·53-s + 55-s + 12·59-s − 2·61-s − 6·65-s − 4·67-s − 6·73-s + 8·79-s + 4·83-s + 85-s + 6·89-s + 4·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 1.66·13-s − 0.242·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s − 0.328·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s − 49-s − 1.92·53-s + 0.134·55-s + 1.56·59-s − 0.256·61-s − 0.744·65-s − 0.488·67-s − 0.702·73-s + 0.900·79-s + 0.439·83-s + 0.108·85-s + 0.635·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.59259928194270, −13.14855859970696, −12.77482662194236, −12.42205421194723, −11.49979251053866, −11.34692725859810, −10.86947560813937, −10.48943784666764, −9.799055768094017, −9.260918189887665, −8.743403223452820, −8.375235097939575, −7.818133111714527, −7.434644097173512, −6.615679475610134, −6.305468147651829, −5.814748620295385, −5.142511184580108, −4.530080444153969, −4.025141348969108, −3.472606663347569, −3.033492473768549, −2.104994811224392, −1.633128195380207, −0.7943527147890476, 0, 0.7943527147890476, 1.633128195380207, 2.104994811224392, 3.033492473768549, 3.472606663347569, 4.025141348969108, 4.530080444153969, 5.142511184580108, 5.814748620295385, 6.305468147651829, 6.615679475610134, 7.434644097173512, 7.818133111714527, 8.375235097939575, 8.743403223452820, 9.260918189887665, 9.799055768094017, 10.48943784666764, 10.86947560813937, 11.34692725859810, 11.49979251053866, 12.42205421194723, 12.77482662194236, 13.14855859970696, 13.59259928194270

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