Properties

Label 2-880-11.3-c1-0-11
Degree $2$
Conductor $880$
Sign $0.513 - 0.857i$
Analytic cond. $7.02683$
Root an. cond. $2.65081$
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  + (0.856 + 2.63i)3-s + (−0.809 − 0.587i)5-s + (1.40 − 4.33i)7-s + (−3.78 + 2.75i)9-s + (3.01 − 1.37i)11-s + (−2.34 + 1.70i)13-s + (0.856 − 2.63i)15-s + (5.73 + 4.16i)17-s + (2.20 + 6.79i)19-s + 12.6·21-s + 1.86·23-s + (0.309 + 0.951i)25-s + (−3.77 − 2.74i)27-s + (−0.312 + 0.961i)29-s + (3.56 − 2.59i)31-s + ⋯
L(s)  = 1  + (0.494 + 1.52i)3-s + (−0.361 − 0.262i)5-s + (0.532 − 1.63i)7-s + (−1.26 + 0.917i)9-s + (0.910 − 0.413i)11-s + (−0.650 + 0.472i)13-s + (0.221 − 0.680i)15-s + (1.39 + 1.01i)17-s + (0.506 + 1.55i)19-s + 2.75·21-s + 0.389·23-s + (0.0618 + 0.190i)25-s + (−0.726 − 0.527i)27-s + (−0.0579 + 0.178i)29-s + (0.640 − 0.465i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.513 - 0.857i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.513 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68290 + 0.953571i\)
\(L(\frac12)\) \(\approx\) \(1.68290 + 0.953571i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-3.01 + 1.37i)T \)
good3 \( 1 + (-0.856 - 2.63i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-1.40 + 4.33i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.34 - 1.70i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.73 - 4.16i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.20 - 6.79i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.86T + 23T^{2} \)
29 \( 1 + (0.312 - 0.961i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.56 + 2.59i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.66 - 5.12i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.38 + 4.26i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.73T + 43T^{2} \)
47 \( 1 + (2.42 + 7.46i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.578 - 0.419i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.17 + 3.61i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.71 + 1.24i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + (-3.40 - 2.47i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.422 + 1.29i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.90 - 7.19i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (13.2 + 9.60i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (-8.78 + 6.38i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.985427030928870018780222118542, −9.848628055276109920009874513313, −8.564466884617937610227388388436, −7.985761235994775760097526591003, −7.04332604070310054053499892894, −5.64646072059966346705059974140, −4.60953752259665972933001363820, −3.87025958290494629017517914541, −3.47049206824992571357646917601, −1.32231655455294919657951104138, 1.11508804754857672434664067609, 2.47057387629162548291068512967, 2.98543124482402521766143157141, 4.86111988394176142294978509482, 5.75059082849041264493495311448, 6.83329660041832889804469639008, 7.43462846802480857213022933639, 8.178771542290934627126561463741, 9.023071733223952382431437739634, 9.648404225617878009587510730393

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