Properties

Label 2-2240-8.5-c1-0-8
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  + 2.89i·3-s + i·5-s − 7-s − 5.38·9-s + 2.38i·11-s + 3.40i·13-s − 2.89·15-s − 1.92·17-s + 5.79i·19-s − 2.89i·21-s + 8.76·23-s − 25-s − 6.89i·27-s + 5.86i·29-s + 1.02·31-s + ⋯
L(s)  = 1  + 1.67i·3-s + 0.447i·5-s − 0.377·7-s − 1.79·9-s + 0.718i·11-s + 0.945i·13-s − 0.747·15-s − 0.466·17-s + 1.32i·19-s − 0.631i·21-s + 1.82·23-s − 0.200·25-s − 1.32i·27-s + 1.08i·29-s + 0.184·31-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: $\chi_{2240} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.120146129\)
\(L(\frac12)\) \(\approx\) \(1.120146129\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.89iT - 3T^{2} \)
11 \( 1 - 2.38iT - 11T^{2} \)
13 \( 1 - 3.40iT - 13T^{2} \)
17 \( 1 + 1.92T + 17T^{2} \)
19 \( 1 - 5.79iT - 19T^{2} \)
23 \( 1 - 8.76T + 23T^{2} \)
29 \( 1 - 5.86iT - 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 + 2.81iT - 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 5.40T + 47T^{2} \)
53 \( 1 + 6.97iT - 53T^{2} \)
59 \( 1 + 6.97iT - 59T^{2} \)
61 \( 1 + 14.7iT - 61T^{2} \)
67 \( 1 + 0.209iT - 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 5.79iT - 83T^{2} \)
89 \( 1 + 1.23T + 89T^{2} \)
97 \( 1 - 5.92T + 97T^{2} \)
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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

−9.581969386488071493380877748011, −9.028795991731956947127092597351, −8.224783978653710534772012667133, −7.00027018009332965958423181382, −6.46287869623071431590862633452, −5.21378218837297838331471344424, −4.78551518890209605662137116457, −3.72490319127573706161840418341, −3.27688119723841179328246677484, −1.94655996479284986221289336963, 0.40992419445077651795444675953, 1.22276126376373067973429259391, 2.56502039389646904231701807476, 3.17571515127938599391635393301, 4.71393121014216437546103253741, 5.58903316374528009200562998566, 6.37245142654450568661600783919, 7.00616873672750776425758811593, 7.68598431234982025472453719674, 8.569865493089745667236457989643

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