Properties

Label 2-880-5.4-c1-0-7
Degree $2$
Conductor $880$
Sign $-0.447 - 0.894i$
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  + 2i·3-s + (1 + 2i)5-s − 4i·7-s − 9-s − 11-s + 6i·13-s + (−4 + 2i)15-s + 2i·17-s + 4·19-s + 8·21-s + 6i·23-s + (−3 + 4i)25-s + 4i·27-s + 2·29-s − 8·31-s + ⋯
L(s)  = 1  + 1.15i·3-s + (0.447 + 0.894i)5-s − 1.51i·7-s − 0.333·9-s − 0.301·11-s + 1.66i·13-s + (−1.03 + 0.516i)15-s + 0.485i·17-s + 0.917·19-s + 1.74·21-s + 1.25i·23-s + (−0.600 + 0.800i)25-s + 0.769i·27-s + 0.371·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.830385 + 1.34359i\)
\(L(\frac12)\) \(\approx\) \(0.830385 + 1.34359i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2i)T \)
11 \( 1 + T \)
good3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 10iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 4iT - 97T^{2} \)
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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

−10.45803462893824144279072245183, −9.573130176757961113313804112957, −9.262215190844250844017593468252, −7.57063300568067126383469153718, −7.14917821214173835529832113229, −6.08342573587508510184956483734, −4.92032197604678558133236977117, −3.96807518765375229986214574327, −3.42194278218698302765357204583, −1.72783014168056431748981072373, 0.78412738756074621289558862968, 2.10531266647646563688591560793, 2.96986766000025597926792122965, 4.87544534309813936394287031326, 5.59469975944140291401404749680, 6.22752483856707988059263737864, 7.48005622337731277326750103828, 8.166757413117946499717744869165, 8.876367017848573481397020461682, 9.693964136396894298368878541269

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