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Properties

Label 2-2240-140.139-c1-0-52
Degree $2$
Conductor $2240$
Sign $0.952 - 0.303i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.568i·3-s + (2.03 + 0.918i)5-s + (1.76 + 1.96i)7-s + 2.67·9-s − 0.417i·11-s + 2.13·13-s + (0.521 − 1.15i)15-s − 0.314·17-s − 2.19·19-s + (1.11 − 1.00i)21-s + 0.992·23-s + (3.31 + 3.74i)25-s − 3.22i·27-s + 4.35·29-s + 4.22·31-s + ⋯
L(s)  = 1  − 0.328i·3-s + (0.911 + 0.410i)5-s + (0.667 + 0.744i)7-s + 0.892·9-s − 0.125i·11-s + 0.592·13-s + (0.134 − 0.299i)15-s − 0.0761·17-s − 0.503·19-s + (0.244 − 0.219i)21-s + 0.206·23-s + (0.662 + 0.748i)25-s − 0.620i·27-s + 0.808·29-s + 0.759·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.952 - 0.303i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.952 - 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.614745205\)
\(L(\frac12)\) \(\approx\) \(2.614745205\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 - 0.918i)T \)
7 \( 1 + (-1.76 - 1.96i)T \)
good3 \( 1 + 0.568iT - 3T^{2} \)
11 \( 1 + 0.417iT - 11T^{2} \)
13 \( 1 - 2.13T + 13T^{2} \)
17 \( 1 + 0.314T + 17T^{2} \)
19 \( 1 + 2.19T + 19T^{2} \)
23 \( 1 - 0.992T + 23T^{2} \)
29 \( 1 - 4.35T + 29T^{2} \)
31 \( 1 - 4.22T + 31T^{2} \)
37 \( 1 + 6.83iT - 37T^{2} \)
41 \( 1 - 3.82iT - 41T^{2} \)
43 \( 1 + 6.04T + 43T^{2} \)
47 \( 1 - 4.98iT - 47T^{2} \)
53 \( 1 + 7.14iT - 53T^{2} \)
59 \( 1 + 9.50T + 59T^{2} \)
61 \( 1 + 0.783iT - 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 6.55iT - 71T^{2} \)
73 \( 1 + 5.11T + 73T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + 5.49iT - 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 + 0.314T + 97T^{2} \)
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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

−9.056029284365312854994975762201, −8.332871925987789570231122072582, −7.53241195892430538967385649497, −6.55586458276564326977822821378, −6.12360192689170048727041426428, −5.14976318671695553239020010855, −4.35359873197959132671905631374, −3.05966298152403138639048920805, −2.10086589017990453212926701085, −1.30200580023995491303112262185, 1.08986168751286482629916315099, 1.87949057645403888569691214243, 3.24723663667080209173639772656, 4.51551109459947621817447070173, 4.66948830783477659855357012291, 5.85958802693959965679791647188, 6.64997371242242881067008585099, 7.41143684288440922581236443416, 8.405357315062448134465841524657, 8.927445923821780886520689854816

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