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Properties

Label 2-880-44.39-c1-0-12
Degree $2$
Conductor $880$
Sign $0.962 - 0.272i$
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  + (1.06 − 1.46i)3-s + (−0.309 + 0.951i)5-s + (−1.73 + 1.26i)7-s + (−0.0808 − 0.248i)9-s + (1.48 + 2.96i)11-s + (1.96 − 0.637i)13-s + (1.06 + 1.46i)15-s + (6.94 + 2.25i)17-s + (−5.12 − 3.72i)19-s + 3.87i·21-s + 1.68i·23-s + (−0.809 − 0.587i)25-s + (4.70 + 1.52i)27-s + (4.84 + 6.66i)29-s + (5.64 − 1.83i)31-s + ⋯
L(s)  = 1  + (0.612 − 0.843i)3-s + (−0.138 + 0.425i)5-s + (−0.656 + 0.476i)7-s + (−0.0269 − 0.0829i)9-s + (0.447 + 0.894i)11-s + (0.544 − 0.176i)13-s + (0.274 + 0.377i)15-s + (1.68 + 0.547i)17-s + (−1.17 − 0.854i)19-s + 0.845i·21-s + 0.351i·23-s + (−0.161 − 0.117i)25-s + (0.905 + 0.294i)27-s + (0.899 + 1.23i)29-s + (1.01 − 0.329i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.962 - 0.272i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.962 - 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83121 + 0.254673i\)
\(L(\frac12)\) \(\approx\) \(1.83121 + 0.254673i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 2.96i)T \)
good3 \( 1 + (-1.06 + 1.46i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (1.73 - 1.26i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.96 + 0.637i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-6.94 - 2.25i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.12 + 3.72i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.68iT - 23T^{2} \)
29 \( 1 + (-4.84 - 6.66i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-5.64 + 1.83i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.33 - 6.05i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.75 + 6.54i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.33T + 43T^{2} \)
47 \( 1 + (2.15 - 2.96i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.00 + 3.09i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.14 + 9.83i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.18 - 0.385i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.89iT - 67T^{2} \)
71 \( 1 + (13.9 + 4.53i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.87 - 8.09i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.285 + 0.879i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.980 + 3.01i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + (-1.54 - 4.75i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.18806779543768142881812619215, −9.212369696587990242161754306366, −8.403011269926109481682053934039, −7.63102032050035704990588396350, −6.79056634195430873144351234103, −6.15497527728382549911352749783, −4.83963630106041678367876667422, −3.50538367117174198837393948798, −2.63395799420684870833999447444, −1.45253964653627573557529710839, 0.951482691202754574656574912061, 2.93804614005146225889808679351, 3.76326398674012464715729495651, 4.40372557747977128773562850417, 5.79740978901291238797342727766, 6.52738355125960601955793402561, 7.82671298176548642098187049940, 8.572309686901921110380363270449, 9.249843841587927649296432467661, 10.11449203003546313350965425799

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