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Properties

Label 2-880-11.9-c1-0-16
Degree $2$
Conductor $880$
Sign $-0.353 + 0.935i$
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  + (−1 + 0.726i)3-s + (−0.309 − 0.951i)5-s + (0.309 + 0.224i)7-s + (−0.454 + 1.40i)9-s + (−3.04 − 1.31i)11-s + (0.190 − 0.587i)13-s + (1 + 0.726i)15-s + (−0.236 − 0.726i)17-s + (5.73 − 4.16i)19-s − 0.472·21-s − 6.85·23-s + (−0.809 + 0.587i)25-s + (−1.70 − 5.25i)27-s + (−2.61 − 1.90i)29-s + (0.854 − 2.62i)31-s + ⋯
L(s)  = 1  + (−0.577 + 0.419i)3-s + (−0.138 − 0.425i)5-s + (0.116 + 0.0848i)7-s + (−0.151 + 0.466i)9-s + (−0.918 − 0.396i)11-s + (0.0529 − 0.163i)13-s + (0.258 + 0.187i)15-s + (−0.0572 − 0.176i)17-s + (1.31 − 0.956i)19-s − 0.103·21-s − 1.42·23-s + (−0.161 + 0.117i)25-s + (−0.328 − 1.01i)27-s + (−0.486 − 0.353i)29-s + (0.153 − 0.472i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.353 + 0.935i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.313266 - 0.453464i\)
\(L(\frac12)\) \(\approx\) \(0.313266 - 0.453464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (3.04 + 1.31i)T \)
good3 \( 1 + (1 - 0.726i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (-0.309 - 0.224i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.190 + 0.587i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.236 + 0.726i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.73 + 4.16i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.85T + 23T^{2} \)
29 \( 1 + (2.61 + 1.90i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.854 + 2.62i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.73 + 3.44i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.92 - 2.85i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.76T + 43T^{2} \)
47 \( 1 + (3.5 - 2.54i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.73 + 11.4i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.73 - 2.71i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.76 + 8.50i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.23T + 67T^{2} \)
71 \( 1 + (2.70 + 8.33i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (9.47 + 6.88i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.23 + 9.95i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.708 + 2.17i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (3.70 - 11.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.01219572598441336972023772673, −9.116438116389528033695976274277, −8.081322818928315232594165276309, −7.55106267863668570943228380312, −6.17219035149514119021436808836, −5.30946780195671178311501907032, −4.80063822540358980346091353767, −3.51607146808153072974886664655, −2.20271126909133908444557902018, −0.28246081713366663937327098389, 1.52077292213956234212611354254, 2.98506324060997226022653458069, 4.05696067116819184744226271208, 5.38476876915232035944229315899, 5.99575938949660192886926574651, 7.06233779471445237453648151727, 7.65103895328449141956117158239, 8.625667159023737051869978547863, 9.780369086880575989062880011547, 10.38039161758816892811926209258

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