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Properties

Label 2-880-11.3-c1-0-4
Degree $2$
Conductor $880$
Sign $-0.941 - 0.335i$
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  + (0.868 + 2.67i)3-s + (0.809 + 0.587i)5-s + (−0.318 + 0.980i)7-s + (−3.96 + 2.88i)9-s + (−1.93 − 2.69i)11-s + (−2.79 + 2.02i)13-s + (−0.868 + 2.67i)15-s + (−1.94 − 1.40i)17-s + (2.36 + 7.29i)19-s − 2.89·21-s − 2.45·23-s + (0.309 + 0.951i)25-s + (−4.33 − 3.14i)27-s + (−1.83 + 5.66i)29-s + (−2.98 + 2.16i)31-s + ⋯
L(s)  = 1  + (0.501 + 1.54i)3-s + (0.361 + 0.262i)5-s + (−0.120 + 0.370i)7-s + (−1.32 + 0.961i)9-s + (−0.583 − 0.811i)11-s + (−0.773 + 0.562i)13-s + (−0.224 + 0.690i)15-s + (−0.470 − 0.341i)17-s + (0.543 + 1.67i)19-s − 0.632·21-s − 0.512·23-s + (0.0618 + 0.190i)25-s + (−0.833 − 0.605i)27-s + (−0.341 + 1.05i)29-s + (−0.535 + 0.389i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.941 - 0.335i$
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: \((2,\ 880,\ (\ :1/2),\ -0.941 - 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251064 + 1.45100i\)
\(L(\frac12)\) \(\approx\) \(0.251064 + 1.45100i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 + 2.69i)T \)
good3 \( 1 + (-0.868 - 2.67i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.318 - 0.980i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.79 - 2.02i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.94 + 1.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.36 - 7.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.45T + 23T^{2} \)
29 \( 1 + (1.83 - 5.66i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.98 - 2.16i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.84 + 5.66i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.21 - 3.74i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.64T + 43T^{2} \)
47 \( 1 + (1.80 + 5.55i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.58 + 6.96i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.910 - 2.80i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.00 + 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.14T + 67T^{2} \)
71 \( 1 + (-1.63 - 1.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.255 - 0.785i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.77 - 7.09i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.30 - 0.946i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.16T + 89T^{2} \)
97 \( 1 + (1.97 - 1.43i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.37011072474786469907015110235, −9.670727629197267511422265890478, −9.062771064219477536108743239902, −8.267113648740554380299992445268, −7.20772259636429414698646626657, −5.81374940264983587276769085223, −5.25777701286262496021250972736, −4.11172224995221627346444682822, −3.27869797593987076232954429755, −2.26810184678963567269061836016, 0.63046740337158463986626242252, 2.10379146800534435647610295489, 2.74678421671360840190480098910, 4.39989256130125828170721038239, 5.52868370431480361192987219402, 6.54458704235128869802665333661, 7.40785715075132343221326848641, 7.73810518220014245695593135342, 8.837954326806085534444966365795, 9.618804272841441442680022123739

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