Properties

Label 2-2240-28.27-c1-0-7
Degree $2$
Conductor $2240$
Sign $-0.611 - 0.791i$
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  + 0.477·3-s i·5-s + (2.09 − 1.61i)7-s − 2.77·9-s + 4.66i·11-s + 3.77i·13-s − 0.477i·15-s + 3.77i·17-s − 7.42·19-s + (1 − 0.772i)21-s − 3.23i·23-s − 25-s − 2.75·27-s − 3.77·29-s − 0.954·31-s + ⋯
L(s)  = 1  + 0.275·3-s − 0.447i·5-s + (0.791 − 0.611i)7-s − 0.924·9-s + 1.40i·11-s + 1.04i·13-s − 0.123i·15-s + 0.914i·17-s − 1.70·19-s + (0.218 − 0.168i)21-s − 0.674i·23-s − 0.200·25-s − 0.530·27-s − 0.700·29-s − 0.171·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.611 - 0.791i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.611 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7955926910\)
\(L(\frac12)\) \(\approx\) \(0.7955926910\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (-2.09 + 1.61i)T \)
good3 \( 1 - 0.477T + 3T^{2} \)
11 \( 1 - 4.66iT - 11T^{2} \)
13 \( 1 - 3.77iT - 13T^{2} \)
17 \( 1 - 3.77iT - 17T^{2} \)
19 \( 1 + 7.42T + 19T^{2} \)
23 \( 1 + 3.23iT - 23T^{2} \)
29 \( 1 + 3.77T + 29T^{2} \)
31 \( 1 + 0.954T + 31T^{2} \)
37 \( 1 + 5.54T + 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 + 7.89T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 9.33T + 59T^{2} \)
61 \( 1 + 13.5iT - 61T^{2} \)
67 \( 1 - 6.09iT - 67T^{2} \)
71 \( 1 - 9.33iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + 8.22iT - 97T^{2} \)
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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

−9.188309240844471515218451126558, −8.453872052701306468377177122368, −7.980989792254326292924024887522, −6.97531333696845349120802301274, −6.31803851397339619561641298316, −5.16536536468675331913516349237, −4.41388529060510590123456110720, −3.85275182712232512506725121441, −2.25966340174798885001726353107, −1.66357356660945894760831464232, 0.24033633716801940210936916783, 1.97298909574946531681187440859, 2.92479101026762195505056649634, 3.56013583478075825578220276875, 4.92312436612485950483565186665, 5.68613036626058232875567490051, 6.19327504624598658402075913779, 7.40018442606084303511725002705, 8.149814545543627085367320636995, 8.676169053181831453812335221286

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