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Properties

Label 2-880-55.7-c1-0-18
Degree $2$
Conductor $880$
Sign $0.618 + 0.785i$
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  + (0.361 + 0.710i)3-s + (−1.89 + 1.18i)5-s + (−0.170 − 0.0869i)7-s + (1.38 − 1.91i)9-s + (−1.77 − 2.80i)11-s + (−0.484 − 3.05i)13-s + (−1.52 − 0.916i)15-s + (0.579 − 3.66i)17-s + (0.229 − 0.707i)19-s − 0.152i·21-s + (1.14 − 1.14i)23-s + (2.17 − 4.49i)25-s + (4.22 + 0.669i)27-s + (2.95 + 9.07i)29-s + (0.283 + 0.206i)31-s + ⋯
L(s)  = 1  + (0.208 + 0.410i)3-s + (−0.847 + 0.531i)5-s + (−0.0644 − 0.0328i)7-s + (0.463 − 0.637i)9-s + (−0.535 − 0.844i)11-s + (−0.134 − 0.847i)13-s + (−0.394 − 0.236i)15-s + (0.140 − 0.887i)17-s + (0.0527 − 0.162i)19-s − 0.0333i·21-s + (0.239 − 0.239i)23-s + (0.435 − 0.899i)25-s + (0.812 + 0.128i)27-s + (0.547 + 1.68i)29-s + (0.0509 + 0.0370i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.618 + 0.785i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.618 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09962 - 0.533871i\)
\(L(\frac12)\) \(\approx\) \(1.09962 - 0.533871i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 - 1.18i)T \)
11 \( 1 + (1.77 + 2.80i)T \)
good3 \( 1 + (-0.361 - 0.710i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (0.170 + 0.0869i)T + (4.11 + 5.66i)T^{2} \)
13 \( 1 + (0.484 + 3.05i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.579 + 3.66i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-0.229 + 0.707i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.14 + 1.14i)T - 23iT^{2} \)
29 \( 1 + (-2.95 - 9.07i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.283 - 0.206i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.45 + 4.81i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-6.36 - 2.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.72 + 3.72i)T + 43iT^{2} \)
47 \( 1 + (-11.0 + 5.61i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (8.91 - 1.41i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (-9.15 + 2.97i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.46 + 4.76i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.13 + 4.13i)T + 67iT^{2} \)
71 \( 1 + (9.27 - 6.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.09 + 2.14i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-0.542 - 0.394i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (16.4 + 2.60i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + 7.92iT - 89T^{2} \)
97 \( 1 + (0.215 + 1.36i)T + (-92.2 + 29.9i)T^{2} \)
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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

−10.18221984596681152834305569142, −9.124974470364701138615429538893, −8.378024013707671173732061356690, −7.43750835569815948661805557009, −6.76834541347815356318941326660, −5.56417376418263589864491682141, −4.55063465553279156978792061881, −3.43124483386771623195648863946, −2.87624174412500478282710848581, −0.61930152207470493889228874494, 1.42919599215958431862578450976, 2.64057426657185060517958102034, 4.20793637321038224009334002605, 4.64255392435706878767827810031, 5.96350563306873639204262245657, 7.11106620308292584384229204809, 7.75489121203734891139466370779, 8.347087358259779952082043950191, 9.413020417035289257949503782643, 10.20665329808999429590294038855

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