Properties

Label 2-2240-1.1-c1-0-31
Degree $2$
Conductor $2240$
Sign $1$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  + 3·3-s + 5-s + 7-s + 6·9-s − 5·11-s + 3·13-s + 3·15-s − 17-s + 6·19-s + 3·21-s − 6·23-s + 25-s + 9·27-s + 9·29-s + 4·31-s − 15·33-s + 35-s − 2·37-s + 9·39-s − 4·41-s + 10·43-s + 6·45-s + 47-s + 49-s − 3·51-s − 4·53-s − 5·55-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 1.50·11-s + 0.832·13-s + 0.774·15-s − 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s + 0.718·31-s − 2.61·33-s + 0.169·35-s − 0.328·37-s + 1.44·39-s − 0.624·41-s + 1.52·43-s + 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s − 0.549·53-s − 0.674·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.805093272\)
\(L(\frac12)\) \(\approx\) \(3.805093272\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.861920653937128870724599342485, −8.213043350946324211142347413159, −7.85081335095312046952512435207, −6.96929537196328486214467340283, −5.86772395707333722143344910386, −4.92379386705551451390155657739, −3.98659668409059060417281652635, −2.96940713374575090223990451771, −2.44270886572835496551406086694, −1.32708580773053885811875668723, 1.32708580773053885811875668723, 2.44270886572835496551406086694, 2.96940713374575090223990451771, 3.98659668409059060417281652635, 4.92379386705551451390155657739, 5.86772395707333722143344910386, 6.96929537196328486214467340283, 7.85081335095312046952512435207, 8.213043350946324211142347413159, 8.861920653937128870724599342485

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