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Properties

Label 2-880-55.43-c1-0-25
Degree $2$
Conductor $880$
Sign $-0.983 + 0.180i$
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  + (−1.15 − 1.15i)3-s + (−1.65 + 1.5i)5-s − 0.316i·9-s + 3.31·11-s + (3.65 + 0.183i)15-s + (−6.15 − 6.15i)23-s + (0.5 − 4.97i)25-s + (−3.84 + 3.84i)27-s − 9.94·31-s + (−3.84 − 3.84i)33-s + (−8.47 + 8.47i)37-s + (0.474 + 0.525i)45-s + (2.68 − 2.68i)47-s − 7i·49-s + (−9.63 − 9.63i)53-s + ⋯
L(s)  = 1  + (−0.668 − 0.668i)3-s + (−0.741 + 0.670i)5-s − 0.105i·9-s + 1.00·11-s + (0.944 + 0.0473i)15-s + (−1.28 − 1.28i)23-s + (0.100 − 0.994i)25-s + (−0.739 + 0.739i)27-s − 1.78·31-s + (−0.668 − 0.668i)33-s + (−1.39 + 1.39i)37-s + (0.0707 + 0.0782i)45-s + (0.391 − 0.391i)47-s − i·49-s + (−1.32 − 1.32i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.983 + 0.180i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.983 + 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0281495 - 0.308924i\)
\(L(\frac12)\) \(\approx\) \(0.0281495 - 0.308924i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 - 1.5i)T \)
11 \( 1 - 3.31T \)
good3 \( 1 + (1.15 + 1.15i)T + 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (6.15 + 6.15i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 9.94T + 31T^{2} \)
37 \( 1 + (8.47 - 8.47i)T - 37iT^{2} \)
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 \( 1 - 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (1.52 - 1.52i)T - 67iT^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (13.4 - 13.4i)T - 97iT^{2} \)
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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

−9.870572331368526365360788858751, −8.782379915621185713693318219014, −7.938008754512073921489585436511, −6.81730525078917028128866720539, −6.64318968508728862006930549438, −5.53827942895238510689481172655, −4.19427755859513955622835854777, −3.36457359205956965991956043735, −1.78734510435826224394131481270, −0.16310798549307217222084499513, 1.66189278652108668475544883597, 3.64657576234424018936121435033, 4.22127008656462026332028694621, 5.25530335625440051681050787391, 5.93739024546715926542855212596, 7.25291785836381129923552371752, 7.958908009250730917736907822717, 9.088798403922710060415718187318, 9.559533857559024292354646545251, 10.75624431114481377447337374623

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