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Properties

Label 2-880-55.32-c1-0-13
Degree $2$
Conductor $880$
Sign $0.648 - 0.761i$
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  + (−1.28 + 1.28i)3-s + (2.12 − 0.686i)5-s − 0.328i·9-s + 3.31·11-s + (−1.86 + 3.63i)15-s + (0.618 − 0.618i)23-s + (4.05 − 2.92i)25-s + (−3.44 − 3.44i)27-s + 9.30·31-s + (−4.27 + 4.27i)33-s + (5.51 + 5.51i)37-s + (−0.225 − 0.698i)45-s + (2.68 + 2.68i)47-s + 7i·49-s + (−9.63 + 9.63i)53-s + ⋯
L(s)  = 1  + (−0.744 + 0.744i)3-s + (0.951 − 0.306i)5-s − 0.109i·9-s + 1.00·11-s + (−0.480 + 0.937i)15-s + (0.128 − 0.128i)23-s + (0.811 − 0.584i)25-s + (−0.663 − 0.663i)27-s + 1.67·31-s + (−0.744 + 0.744i)33-s + (0.906 + 0.906i)37-s + (−0.0335 − 0.104i)45-s + (0.391 + 0.391i)47-s + i·49-s + (−1.32 + 1.32i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.648 - 0.761i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38292 + 0.638810i\)
\(L(\frac12)\) \(\approx\) \(1.38292 + 0.638810i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.686i)T \)
11 \( 1 - 3.31T \)
good3 \( 1 + (1.28 - 1.28i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-0.618 + 0.618i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 9.30T + 31T^{2} \)
37 \( 1 + (-5.51 - 5.51i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-2.68 - 2.68i)T + 47iT^{2} \)
53 \( 1 + (9.63 - 9.63i)T - 53iT^{2} \)
59 \( 1 - 14.6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (9.17 + 9.17i)T + 67iT^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 18.8iT - 89T^{2} \)
97 \( 1 + (-9.79 - 9.79i)T + 97iT^{2} \)
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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

−10.22189075838181633924132296248, −9.554817186850275807237180944739, −8.863036675970693958791326544355, −7.74964070676388170103987296216, −6.39238383666055405996141096062, −5.98690099848352018462891345734, −4.85910012752428332128794917300, −4.29992990994630577179947662781, −2.75710457416585598054801325831, −1.26236033486167586396059832514, 0.990721164366009958653990543871, 2.14425333866750053104633949067, 3.54914160267085133115080878504, 4.92093074809775368618685517862, 5.94474371100048568672504200613, 6.49301631789935072169410149629, 7.12570789384268802271630609018, 8.326047384763977679113903659503, 9.371654707750414584927849408837, 9.932901483755333747187063984830

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