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Properties

Label 2-880-55.28-c1-0-26
Degree $2$
Conductor $880$
Sign $0.0484 + 0.998i$
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  + (0.0342 + 0.216i)3-s + (−0.388 − 2.20i)5-s + (−0.104 − 0.0165i)7-s + (2.80 − 0.912i)9-s + (0.733 − 3.23i)11-s + (1.77 + 0.904i)13-s + (0.463 − 0.159i)15-s + (−4.77 + 2.43i)17-s + (−3.72 − 2.70i)19-s − 0.0232i·21-s + (1.26 + 1.26i)23-s + (−4.69 + 1.71i)25-s + (0.592 + 1.16i)27-s + (3.14 − 2.28i)29-s + (−2.80 − 8.62i)31-s + ⋯
L(s)  = 1  + (0.0197 + 0.124i)3-s + (−0.173 − 0.984i)5-s + (−0.0396 − 0.00627i)7-s + (0.935 − 0.304i)9-s + (0.221 − 0.975i)11-s + (0.492 + 0.250i)13-s + (0.119 − 0.0412i)15-s + (−1.15 + 0.590i)17-s + (−0.855 − 0.621i)19-s − 0.00507i·21-s + (0.263 + 0.263i)23-s + (−0.939 + 0.342i)25-s + (0.113 + 0.223i)27-s + (0.583 − 0.423i)29-s + (−0.503 − 1.54i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0484 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0484 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.0484 + 0.998i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.0484 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02967 - 0.980910i\)
\(L(\frac12)\) \(\approx\) \(1.02967 - 0.980910i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.388 + 2.20i)T \)
11 \( 1 + (-0.733 + 3.23i)T \)
good3 \( 1 + (-0.0342 - 0.216i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (0.104 + 0.0165i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (-1.77 - 0.904i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (4.77 - 2.43i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (3.72 + 2.70i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.26 - 1.26i)T + 23iT^{2} \)
29 \( 1 + (-3.14 + 2.28i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.80 + 8.62i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.42 + 9.02i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-0.361 + 0.498i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-2.10 + 2.10i)T - 43iT^{2} \)
47 \( 1 + (-8.76 + 1.38i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-3.65 + 7.17i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-6.35 - 8.75i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.692 + 0.225i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.43 - 1.43i)T - 67iT^{2} \)
71 \( 1 + (0.219 - 0.674i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.08 - 13.1i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (2.31 + 7.12i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.58 - 5.07i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 0.464iT - 89T^{2} \)
97 \( 1 + (1.58 + 0.809i)T + (57.0 + 78.4i)T^{2} \)
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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

−9.804923355616576193656240288955, −8.911805004366803089756669978994, −8.565884999463652229443617585863, −7.40609103903363582840970728825, −6.42084086906073624506305999287, −5.57748970898845019253470714267, −4.26984727073587285504415114099, −3.92674176242326444755163366853, −2.12934756127681757883596082364, −0.70552379528220583721263248010, 1.68848917039983804363998559427, 2.83645805869843121939343867986, 4.07668284137183188554281942976, 4.86104847542611746966628628797, 6.37858942316404650015720957874, 6.85991695645786005885224776983, 7.63625333789410134219588188284, 8.630095641565150557315373181707, 9.647234292606696747223652621249, 10.48630716905484350475691936636

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