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Properties

Label 2-880-11.3-c1-0-23
Degree $2$
Conductor $880$
Sign $-0.259 - 0.965i$
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  + (−0.952 − 2.93i)3-s + (−0.809 − 0.587i)5-s + (−0.0119 + 0.0366i)7-s + (−5.25 + 3.81i)9-s + (1.11 − 3.12i)11-s + (−2.79 + 2.02i)13-s + (−0.952 + 2.93i)15-s + (−0.726 − 0.527i)17-s + (−0.373 − 1.14i)19-s + 0.118·21-s − 7.60·23-s + (0.309 + 0.951i)25-s + (8.71 + 6.33i)27-s + (−2.33 + 7.19i)29-s + (−4.81 + 3.50i)31-s + ⋯
L(s)  = 1  + (−0.549 − 1.69i)3-s + (−0.361 − 0.262i)5-s + (−0.00450 + 0.0138i)7-s + (−1.75 + 1.27i)9-s + (0.335 − 0.942i)11-s + (−0.774 + 0.562i)13-s + (−0.245 + 0.756i)15-s + (−0.176 − 0.127i)17-s + (−0.0855 − 0.263i)19-s + 0.0259·21-s − 1.58·23-s + (0.0618 + 0.190i)25-s + (1.67 + 1.21i)27-s + (−0.433 + 1.33i)29-s + (−0.865 + 0.628i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.259 - 0.965i$
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: \((2,\ 880,\ (\ :1/2),\ -0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.125466 + 0.163649i\)
\(L(\frac12)\) \(\approx\) \(0.125466 + 0.163649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 3.12i)T \)
good3 \( 1 + (0.952 + 2.93i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.0119 - 0.0366i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.79 - 2.02i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.726 + 0.527i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.373 + 1.14i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.60T + 23T^{2} \)
29 \( 1 + (2.33 - 7.19i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.81 - 3.50i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.31 + 10.1i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.954 - 2.93i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.68T + 43T^{2} \)
47 \( 1 + (-0.403 - 1.24i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.64 - 1.19i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.0573 + 0.176i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.91 - 5.74i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 8.10T + 67T^{2} \)
71 \( 1 + (5.81 + 4.22i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.92 + 12.0i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.2 - 7.44i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.69 + 6.31i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.78T + 89T^{2} \)
97 \( 1 + (14.3 - 10.4i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.282689665860721005572851252878, −8.555458208621525372841061879170, −7.56541535212102738521594992256, −7.12888696588134924034890920407, −6.10084784718361630770407259626, −5.47996562997889440079636291163, −4.11391611650145238586592619405, −2.60982137500247868389276070025, −1.46246299516013174752177307504, −0.10555039324007767530541063918, 2.49887652043534406384532308254, 3.92529837754976890383084669212, 4.28698035753948619857420773281, 5.37997198961931566176179767156, 6.14692426768127628210243259682, 7.38297366461981143627644533520, 8.319377081338024692227544422758, 9.487362733273115243243028833405, 9.928566909329621418113133225065, 10.50420070982577201853027315001

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