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Properties

Label 2-2240-8.5-c1-0-40
Degree $2$
Conductor $2240$
Sign $-0.258 + 0.965i$
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  + 2.11i·3-s − i·5-s − 7-s − 1.48·9-s − 2.75i·11-s + 3.19i·13-s + 2.11·15-s + 0.352·17-s − 8.13i·19-s − 2.11i·21-s − 7.61·23-s − 25-s + 3.21i·27-s + 2.84i·29-s − 8.60·31-s + ⋯
L(s)  = 1  + 1.22i·3-s − 0.447i·5-s − 0.377·7-s − 0.493·9-s − 0.831i·11-s + 0.886i·13-s + 0.546·15-s + 0.0856·17-s − 1.86i·19-s − 0.461i·21-s − 1.58·23-s − 0.200·25-s + 0.618i·27-s + 0.528i·29-s − 1.54·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4261670798\)
\(L(\frac12)\) \(\approx\) \(0.4261670798\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 2.11iT - 3T^{2} \)
11 \( 1 + 2.75iT - 11T^{2} \)
13 \( 1 - 3.19iT - 13T^{2} \)
17 \( 1 - 0.352T + 17T^{2} \)
19 \( 1 + 8.13iT - 19T^{2} \)
23 \( 1 + 7.61T + 23T^{2} \)
29 \( 1 - 2.84iT - 29T^{2} \)
31 \( 1 + 8.60T + 31T^{2} \)
37 \( 1 - 8.38iT - 37T^{2} \)
41 \( 1 + 6.89T + 41T^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 + 4.41T + 47T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 + 13.3iT - 59T^{2} \)
61 \( 1 + 4.55iT - 61T^{2} \)
67 \( 1 - 4.63iT - 67T^{2} \)
71 \( 1 - 1.00T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 4.60T + 79T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 + 5.61T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−8.855122919188444574139615038451, −8.439604846244880015447517086182, −7.15592670490847048386325724727, −6.45048416201587581947299663251, −5.36341813808004693648438659280, −4.81902325988370180396712442925, −3.88995857840450541156143650812, −3.28205384032227798175494658822, −1.90086563419011620837543936406, −0.13948470951571052827378261389, 1.49573992640798450970679494603, 2.25396592373702718006659203001, 3.40732432161905386939129178946, 4.30103262601030202469324389730, 5.82889219461691999346961254923, 6.01641613644279332507269015190, 7.11033402488208277552398917621, 7.67632601610117555819915666462, 8.085688285913719493686759678067, 9.280949549289207976911492710437

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