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Properties

Label 2-2240-280.139-c1-0-12
Degree $2$
Conductor $2240$
Sign $-0.707 - 0.707i$
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  + 2.23·3-s − 2.23i·5-s + 2.64i·7-s + 2.00·9-s − 5.91·11-s + 6.70i·13-s − 5.00i·15-s − 7.93·17-s + 5.91i·21-s − 5.00·25-s − 2.23·27-s + 5.91i·29-s − 13.2·33-s + 5.91·35-s + 15.0i·39-s + ⋯
L(s)  = 1  + 1.29·3-s − 0.999i·5-s + 0.999i·7-s + 0.666·9-s − 1.78·11-s + 1.86i·13-s − 1.29i·15-s − 1.92·17-s + 1.29i·21-s − 1.00·25-s − 0.430·27-s + 1.09i·29-s − 2.30·33-s + 0.999·35-s + 2.40i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9406871171\)
\(L(\frac12)\) \(\approx\) \(0.9406871171\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
7 \( 1 - 2.64iT \)
good3 \( 1 - 2.23T + 3T^{2} \)
11 \( 1 + 5.91T + 11T^{2} \)
13 \( 1 - 6.70iT - 13T^{2} \)
17 \( 1 + 7.93T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 5.91iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 7.93iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - iT - 79T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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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

−9.026997456780551877530505248505, −8.718584496908912701444359591948, −8.104007193228701071650402454191, −7.19655578293342051105649219138, −6.22352618476839327305567298072, −5.10132887155818599635423547989, −4.60118499367203180206706227109, −3.49665296362969742802691746861, −2.21028626067719240743543119193, −2.08960502542169951764851859556, 0.23364992927560720929404305971, 2.28701676033102346857554952893, 2.77762166637403075207689719691, 3.54045393434257265308600038505, 4.51764721285069530089488249133, 5.61438565493120470236967487149, 6.60528445978758515483239605479, 7.60675513670297866856269684936, 7.81547164087373928978092147240, 8.503501054912774982020301881217

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