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Properties

Label 2-880-880.109-c0-0-2
Degree $2$
Conductor $880$
Sign $0.923 - 0.382i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.707 + 0.707i)5-s − 1.84i·7-s + (0.923 − 0.382i)8-s + i·9-s + (−0.382 − 0.923i)10-s + (0.707 − 0.707i)11-s + (−0.541 − 0.541i)13-s + (1.70 + 0.707i)14-s + i·16-s + 1.84·17-s + (−0.923 − 0.382i)18-s + 20-s + (0.382 + 0.923i)22-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.707 + 0.707i)5-s − 1.84i·7-s + (0.923 − 0.382i)8-s + i·9-s + (−0.382 − 0.923i)10-s + (0.707 − 0.707i)11-s + (−0.541 − 0.541i)13-s + (1.70 + 0.707i)14-s + i·16-s + 1.84·17-s + (−0.923 − 0.382i)18-s + 20-s + (0.382 + 0.923i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7120506501\)
\(L(\frac12)\) \(\approx\) \(0.7120506501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 - iT^{2} \)
7 \( 1 + 1.84iT - T^{2} \)
13 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
17 \( 1 - 1.84T + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1 + i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.765iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.29752154388522028406835799432, −9.776678140510424837112753940838, −8.151651182168801929461394055838, −7.87689761170313445718726770200, −7.15577734342217511917470174196, −6.38263986196593042949494939888, −5.16024065228519847963263106007, −4.18940969103091778322272049377, −3.29061130509206823617740344142, −0.995039889397388829472766082363, 1.39683265301736417181630808059, 2.75031483247032031675647151043, 3.77537645517098113868996397896, 4.79997127995778685458258412799, 5.73103890918529311385509401574, 7.06393406803218427529422938434, 8.175342620734923029777182036544, 8.780695315834252777662822812177, 9.497754470207648709277344520023, 9.951794230634395031984996597377

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