↑

Properties

Label 2-2240-140.139-c1-0-13
Degree $2$
Conductor $2240$
Sign $-0.980 + 0.194i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.21i·3-s + (1.81 − 1.30i)5-s + (−1.09 + 2.41i)7-s − 1.90·9-s + 0.661i·11-s − 0.235·13-s + (2.88 + 4.02i)15-s − 6.02·17-s − 7.98·19-s + (−5.33 − 2.41i)21-s + 8.48·23-s + (1.61 − 4.73i)25-s + 2.43i·27-s − 7.86·29-s − 1.36·31-s + ⋯
L(s)  = 1  + 1.27i·3-s + (0.813 − 0.581i)5-s + (−0.412 + 0.910i)7-s − 0.633·9-s + 0.199i·11-s − 0.0652·13-s + (0.743 + 1.03i)15-s − 1.46·17-s − 1.83·19-s + (−1.16 − 0.527i)21-s + 1.76·23-s + (0.322 − 0.946i)25-s + 0.468i·27-s − 1.46·29-s − 0.245·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.980 + 0.194i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.980 + 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9004745023\)
\(L(\frac12)\) \(\approx\) \(0.9004745023\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 1.30i)T \)
7 \( 1 + (1.09 - 2.41i)T \)
good3 \( 1 - 2.21iT - 3T^{2} \)
11 \( 1 - 0.661iT - 11T^{2} \)
13 \( 1 + 0.235T + 13T^{2} \)
17 \( 1 + 6.02T + 17T^{2} \)
19 \( 1 + 7.98T + 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 + 1.36T + 31T^{2} \)
37 \( 1 - 6.09iT - 37T^{2} \)
41 \( 1 - 2.79iT - 41T^{2} \)
43 \( 1 + 9.54T + 43T^{2} \)
47 \( 1 - 7.72iT - 47T^{2} \)
53 \( 1 + 8.45iT - 53T^{2} \)
59 \( 1 - 0.929T + 59T^{2} \)
61 \( 1 - 2.28iT - 61T^{2} \)
67 \( 1 + 3.19T + 67T^{2} \)
71 \( 1 + 0.619iT - 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 7.01iT - 79T^{2} \)
83 \( 1 - 7.05iT - 83T^{2} \)
89 \( 1 - 9.97iT - 89T^{2} \)
97 \( 1 + 6.02T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.443457967805640047076794572481, −8.856214551256192959221301141041, −8.408645998897888221687598291598, −6.82178981456783004199590972921, −6.26980509269703514723788367690, −5.20083641788678971472070551575, −4.81148753286257265854335077396, −3.90849064164830333090794038755, −2.75568939341993981243634622714, −1.84366825065115798130061797102, 0.27921517060500573172037250061, 1.70525210983123589163582325305, 2.36724050726574220817499980505, 3.52353266584628378475456932026, 4.59719366372706607717070878405, 5.79357306687583878958701721963, 6.61064853045796104930687352198, 6.90358671247995359798410048089, 7.53754323850376016382183177534, 8.673178704714639091198955293899

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