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Properties

Label 2-2240-140.139-c1-0-0
Degree $2$
Conductor $2240$
Sign $-0.432 + 0.901i$
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  + 1.86i·3-s + (−2.07 − 0.836i)5-s + (−2.64 − 0.168i)7-s − 0.486·9-s + 4.26i·11-s + 5.57·13-s + (1.56 − 3.87i)15-s − 3.01·17-s − 0.0464·19-s + (0.314 − 4.93i)21-s − 5.58·23-s + (3.60 + 3.46i)25-s + 4.69i·27-s − 1.21·29-s − 4.62·31-s + ⋯
L(s)  = 1  + 1.07i·3-s + (−0.927 − 0.374i)5-s + (−0.997 − 0.0635i)7-s − 0.162·9-s + 1.28i·11-s + 1.54·13-s + (0.403 − 0.999i)15-s − 0.731·17-s − 0.0106·19-s + (0.0685 − 1.07i)21-s − 1.16·23-s + (0.720 + 0.693i)25-s + 0.903i·27-s − 0.224·29-s − 0.829·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.432 + 0.901i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.432 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04019186312\)
\(L(\frac12)\) \(\approx\) \(0.04019186312\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 0.836i)T \)
7 \( 1 + (2.64 + 0.168i)T \)
good3 \( 1 - 1.86iT - 3T^{2} \)
11 \( 1 - 4.26iT - 11T^{2} \)
13 \( 1 - 5.57T + 13T^{2} \)
17 \( 1 + 3.01T + 17T^{2} \)
19 \( 1 + 0.0464T + 19T^{2} \)
23 \( 1 + 5.58T + 23T^{2} \)
29 \( 1 + 1.21T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 9.87iT - 37T^{2} \)
41 \( 1 + 10.5iT - 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + 6.78iT - 47T^{2} \)
53 \( 1 + 4.99iT - 53T^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
61 \( 1 + 11.5iT - 61T^{2} \)
67 \( 1 - 6.65T + 67T^{2} \)
71 \( 1 + 5.11iT - 71T^{2} \)
73 \( 1 + 9.67T + 73T^{2} \)
79 \( 1 - 5.90iT - 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + 1.51iT - 89T^{2} \)
97 \( 1 + 3.01T + 97T^{2} \)
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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

−9.651006995356961260929737071601, −8.823846427457093826007315794032, −8.259311279698266311991719334679, −7.12137905070594220830484409398, −6.58258934411651229938372730181, −5.41734042704579677718356257882, −4.56317944073985855499981642918, −3.82093822371683362445087350127, −3.44860011231901575297531331411, −1.79446768909155131044985821045, 0.01516549269064508308380840342, 1.19191356386979170637138628017, 2.58369476678609169044916498627, 3.55817718420879077468464943392, 4.09205017241889321940425734098, 5.75991707114028522349805249340, 6.31566171217408514912840868388, 6.87011961009094120902962176016, 7.74705290298882534731019487567, 8.389006989646739826398461730483

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