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Properties

Label 2-880-55.7-c1-0-21
Degree $2$
Conductor $880$
Sign $0.934 + 0.354i$
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  + (0.271 + 0.532i)3-s + (1.85 + 1.25i)5-s + (−2.30 − 1.17i)7-s + (1.55 − 2.13i)9-s + (2.55 − 2.11i)11-s + (−0.439 − 2.77i)13-s + (−0.166 + 1.32i)15-s + (0.147 − 0.932i)17-s + (1.28 − 3.94i)19-s − 1.54i·21-s + (−0.104 + 0.104i)23-s + (1.84 + 4.64i)25-s + (3.33 + 0.527i)27-s + (−2.14 − 6.60i)29-s + (7.33 + 5.32i)31-s + ⋯
L(s)  = 1  + (0.156 + 0.307i)3-s + (0.827 + 0.561i)5-s + (−0.869 − 0.443i)7-s + (0.517 − 0.712i)9-s + (0.770 − 0.637i)11-s + (−0.121 − 0.769i)13-s + (−0.0429 + 0.342i)15-s + (0.0358 − 0.226i)17-s + (0.294 − 0.905i)19-s − 0.336i·21-s + (−0.0218 + 0.0218i)23-s + (0.369 + 0.929i)25-s + (0.641 + 0.101i)27-s + (−0.398 − 1.22i)29-s + (1.31 + 0.956i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.934 + 0.354i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.934 + 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82891 - 0.335496i\)
\(L(\frac12)\) \(\approx\) \(1.82891 - 0.335496i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.85 - 1.25i)T \)
11 \( 1 + (-2.55 + 2.11i)T \)
good3 \( 1 + (-0.271 - 0.532i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (2.30 + 1.17i)T + (4.11 + 5.66i)T^{2} \)
13 \( 1 + (0.439 + 2.77i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.147 + 0.932i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-1.28 + 3.94i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.104 - 0.104i)T - 23iT^{2} \)
29 \( 1 + (2.14 + 6.60i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.33 - 5.32i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.44 - 6.77i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-3.27 - 1.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.91 - 3.91i)T + 43iT^{2} \)
47 \( 1 + (-0.942 + 0.479i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (4.07 - 0.644i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (6.16 - 2.00i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.59 - 7.70i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.94 + 2.94i)T + 67iT^{2} \)
71 \( 1 + (-1.02 + 0.744i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.804 + 1.57i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (3.51 + 2.55i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.18 - 1.29i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 - 4.23iT - 89T^{2} \)
97 \( 1 + (2.25 + 14.2i)T + (-92.2 + 29.9i)T^{2} \)
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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

−9.909396801759798227100845159940, −9.534360411101515840536508219961, −8.629787682183725460041684562634, −7.32904806756723823061731625091, −6.54149155727146798284258568852, −6.00228490991905174178756187336, −4.67575582418984191605080290076, −3.47943665762547515668617506692, −2.83026916687225696611725295571, −0.991448906204383535831791934314, 1.50461485241400226653197059736, 2.37798916491432724557260914229, 3.89278392913366426482735593979, 4.90728456859049362923433214180, 5.94709789959825285153531050906, 6.68189478743326322225635007380, 7.58445098997758975567637516439, 8.682353267417421293642667978204, 9.445391800499951756099111095548, 9.908006185524590446626118173228

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