Properties

Label 2-2240-5.4-c1-0-53
Degree $2$
Conductor $2240$
Sign $0.677 + 0.735i$
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  + 2.19i·3-s + (−1.64 + 1.51i)5-s + i·7-s − 1.83·9-s + 1.37·11-s − 2.74i·13-s + (−3.32 − 3.61i)15-s − 6.94i·17-s + 1.29·19-s − 2.19·21-s − 8.31i·23-s + (0.412 − 4.98i)25-s + 2.56i·27-s − 8.40·29-s − 9.49·31-s + ⋯
L(s)  = 1  + 1.26i·3-s + (−0.735 + 0.677i)5-s + 0.377i·7-s − 0.610·9-s + 0.414·11-s − 0.760i·13-s + (−0.859 − 0.933i)15-s − 1.68i·17-s + 0.296·19-s − 0.479·21-s − 1.73i·23-s + (0.0825 − 0.996i)25-s + 0.494i·27-s − 1.56·29-s − 1.70·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.677 + 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7004279110\)
\(L(\frac12)\) \(\approx\) \(0.7004279110\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 1.51i)T \)
7 \( 1 - iT \)
good3 \( 1 - 2.19iT - 3T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 + 2.74iT - 13T^{2} \)
17 \( 1 + 6.94iT - 17T^{2} \)
19 \( 1 - 1.29T + 19T^{2} \)
23 \( 1 + 8.31iT - 23T^{2} \)
29 \( 1 + 8.40T + 29T^{2} \)
31 \( 1 + 9.49T + 31T^{2} \)
37 \( 1 - 1.73iT - 37T^{2} \)
41 \( 1 + 5.30T + 41T^{2} \)
43 \( 1 + 7.83iT - 43T^{2} \)
47 \( 1 + 3.48iT - 47T^{2} \)
53 \( 1 + 6.13iT - 53T^{2} \)
59 \( 1 - 6.26T + 59T^{2} \)
61 \( 1 + 6.59T + 61T^{2} \)
67 \( 1 - 1.66iT - 67T^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 - 2.13iT - 73T^{2} \)
79 \( 1 - 5.45T + 79T^{2} \)
83 \( 1 + 2.54iT - 83T^{2} \)
89 \( 1 + 1.43T + 89T^{2} \)
97 \( 1 - 9.69iT - 97T^{2} \)
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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

−9.092632540550878851391409507008, −8.294234661185697684274542247784, −7.32415861538195351751065260462, −6.74276164978804470416466496039, −5.47503926270846591995834143130, −4.94506574643281477968128569425, −3.89821634569635822428741865234, −3.37097514144560132824250793252, −2.37480948731103598343784004860, −0.24824423486203386063120361449, 1.35161925946306966649220961830, 1.79746319196463507261645738383, 3.58908366439039129434071201613, 4.05411861795406416294081119160, 5.31198664060375715622025942249, 6.12647943674534126184646479119, 7.01913102104832862008689715830, 7.60294154555472733375769073620, 8.093423129038237847921783665306, 9.064413552047238055644992928040

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