Properties

Label 2-2240-1.1-c1-0-17
Degree $2$
Conductor $2240$
Sign $1$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·3-s − 5-s + 7-s − 0.561·9-s − 1.56·11-s − 6.68·13-s − 1.56·15-s + 7.56·17-s + 7.12·19-s + 1.56·21-s + 3.12·23-s + 25-s − 5.56·27-s − 0.438·29-s + 6.24·31-s − 2.43·33-s − 35-s + 8.24·37-s − 10.4·39-s − 1.12·41-s + 7.12·43-s + 0.561·45-s + 2.43·47-s + 49-s + 11.8·51-s + 13.1·53-s + 1.56·55-s + ⋯
L(s)  = 1  + 0.901·3-s − 0.447·5-s + 0.377·7-s − 0.187·9-s − 0.470·11-s − 1.85·13-s − 0.403·15-s + 1.83·17-s + 1.63·19-s + 0.340·21-s + 0.651·23-s + 0.200·25-s − 1.07·27-s − 0.0814·29-s + 1.12·31-s − 0.424·33-s − 0.169·35-s + 1.35·37-s − 1.67·39-s − 0.175·41-s + 1.08·43-s + 0.0837·45-s + 0.355·47-s + 0.142·49-s + 1.65·51-s + 1.80·53-s + 0.210·55-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: no
Arithmetic: yes
Character: $\chi_{2240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.215114971\)
\(L(\frac12)\) \(\approx\) \(2.215114971\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 3T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 + 6.68T + 13T^{2} \)
17 \( 1 - 7.56T + 17T^{2} \)
19 \( 1 - 7.12T + 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 + 0.438T + 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 + 1.12T + 41T^{2} \)
43 \( 1 - 7.12T + 43T^{2} \)
47 \( 1 - 2.43T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 6.87T + 61T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 4.24T + 73T^{2} \)
79 \( 1 - 0.684T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 5.12T + 89T^{2} \)
97 \( 1 - 1.31T + 97T^{2} \)
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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

−9.033380050978804627668454502075, −8.037678349064237585599297107016, −7.66276431483655092380456668445, −7.14688642410786182131388040064, −5.60217909648394272337989738456, −5.14687666048628455347750689563, −4.05638536675559219564201833078, −2.96326989046568118604550731557, −2.57566179652403389699986079062, −0.944072251671515634703874090699, 0.944072251671515634703874090699, 2.57566179652403389699986079062, 2.96326989046568118604550731557, 4.05638536675559219564201833078, 5.14687666048628455347750689563, 5.60217909648394272337989738456, 7.14688642410786182131388040064, 7.66276431483655092380456668445, 8.037678349064237585599297107016, 9.033380050978804627668454502075

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