Properties

Label 2-40560-1.1-c1-0-17
Degree $2$
Conductor $40560$
Sign $1$
Analytic cond. $323.873$
Root an. cond. $17.9964$
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  − 3-s − 5-s + 5·7-s + 9-s + 3·11-s + 15-s − 8·17-s + 5·19-s − 5·21-s + 4·23-s + 25-s − 27-s − 4·29-s + 2·31-s − 3·33-s − 5·35-s − 7·37-s + 6·41-s − 6·43-s − 45-s + 3·47-s + 18·49-s + 8·51-s + 53-s − 3·55-s − 5·57-s − 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.88·7-s + 1/3·9-s + 0.904·11-s + 0.258·15-s − 1.94·17-s + 1.14·19-s − 1.09·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 0.359·31-s − 0.522·33-s − 0.845·35-s − 1.15·37-s + 0.937·41-s − 0.914·43-s − 0.149·45-s + 0.437·47-s + 18/7·49-s + 1.12·51-s + 0.137·53-s − 0.404·55-s − 0.662·57-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(323.873\)
Root analytic conductor: \(17.9964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 40560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.264768811\)
\(L(\frac12)\) \(\approx\) \(2.264768811\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 \)
good7 \( 1 - 5 T + p T^{2} \) 1.7.af
11 \( 1 - 3 T + p T^{2} \) 1.11.ad
17 \( 1 + 8 T + p T^{2} \) 1.17.i
19 \( 1 - 5 T + p T^{2} \) 1.19.af
23 \( 1 - 4 T + p T^{2} \) 1.23.ae
29 \( 1 + 4 T + p T^{2} \) 1.29.e
31 \( 1 - 2 T + p T^{2} \) 1.31.ac
37 \( 1 + 7 T + p T^{2} \) 1.37.h
41 \( 1 - 6 T + p T^{2} \) 1.41.ag
43 \( 1 + 6 T + p T^{2} \) 1.43.g
47 \( 1 - 3 T + p T^{2} \) 1.47.ad
53 \( 1 - T + p T^{2} \) 1.53.ab
59 \( 1 + 12 T + p T^{2} \) 1.59.m
61 \( 1 - 2 T + p T^{2} \) 1.61.ac
67 \( 1 + 8 T + p T^{2} \) 1.67.i
71 \( 1 + 2 T + p T^{2} \) 1.71.c
73 \( 1 + p T^{2} \) 1.73.a
79 \( 1 - 2 T + p T^{2} \) 1.79.ac
83 \( 1 + 8 T + p T^{2} \) 1.83.i
89 \( 1 + 11 T + p T^{2} \) 1.89.l
97 \( 1 + p T^{2} \) 1.97.a
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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

−14.90415090556338, −14.14570959767815, −13.88884580879317, −13.27048040937971, −12.53827319126520, −11.98412225677989, −11.49584349128654, −11.22181960098213, −10.87312331013302, −10.20235605977252, −9.304632570303814, −8.885434321145141, −8.453918693309293, −7.711196522077003, −7.205591444056240, −6.823198871318719, −5.978825637426080, −5.379551296835601, −4.674686371255166, −4.498270503499701, −3.781659923709936, −2.878175153008610, −1.882837775694213, −1.468439798510843, −0.5947481620427121, 0.5947481620427121, 1.468439798510843, 1.882837775694213, 2.878175153008610, 3.781659923709936, 4.498270503499701, 4.674686371255166, 5.379551296835601, 5.978825637426080, 6.823198871318719, 7.205591444056240, 7.711196522077003, 8.453918693309293, 8.885434321145141, 9.304632570303814, 10.20235605977252, 10.87312331013302, 11.22181960098213, 11.49584349128654, 11.98412225677989, 12.53827319126520, 13.27048040937971, 13.88884580879317, 14.14570959767815, 14.90415090556338

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