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Properties

Label 2-884-884.787-c0-0-0
Degree $2$
Conductor $884$
Sign $0.614 - 0.788i$
Analytic cond. $0.441173$
Root an. cond. $0.664208$
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  + (0.608 + 0.793i)2-s + (−0.258 + 0.965i)4-s + (1.09 − 0.732i)5-s + (−0.923 + 0.382i)8-s + (0.991 − 0.130i)9-s + (1.24 + 0.423i)10-s + (−0.608 − 0.793i)13-s + (−0.866 − 0.499i)16-s + (−0.965 + 0.258i)17-s + (0.707 + 0.707i)18-s + (0.423 + 1.24i)20-s + (0.282 − 0.682i)25-s + (0.258 − 0.965i)26-s + (−0.423 + 0.483i)29-s + (−0.130 − 0.991i)32-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)2-s + (−0.258 + 0.965i)4-s + (1.09 − 0.732i)5-s + (−0.923 + 0.382i)8-s + (0.991 − 0.130i)9-s + (1.24 + 0.423i)10-s + (−0.608 − 0.793i)13-s + (−0.866 − 0.499i)16-s + (−0.965 + 0.258i)17-s + (0.707 + 0.707i)18-s + (0.423 + 1.24i)20-s + (0.282 − 0.682i)25-s + (0.258 − 0.965i)26-s + (−0.423 + 0.483i)29-s + (−0.130 − 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $0.614 - 0.788i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ 0.614 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.520978709\)
\(L(\frac12)\) \(\approx\) \(1.520978709\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 - 0.793i)T \)
13 \( 1 + (0.608 + 0.793i)T \)
17 \( 1 + (0.965 - 0.258i)T \)
good3 \( 1 + (-0.991 + 0.130i)T^{2} \)
5 \( 1 + (-1.09 + 0.732i)T + (0.382 - 0.923i)T^{2} \)
7 \( 1 + (0.991 + 0.130i)T^{2} \)
11 \( 1 + (0.793 + 0.608i)T^{2} \)
19 \( 1 + (-0.965 + 0.258i)T^{2} \)
23 \( 1 + (-0.608 + 0.793i)T^{2} \)
29 \( 1 + (0.423 - 0.483i)T + (-0.130 - 0.991i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (1.31 - 1.50i)T + (-0.130 - 0.991i)T^{2} \)
41 \( 1 + (-0.793 - 1.60i)T + (-0.608 + 0.793i)T^{2} \)
43 \( 1 + (0.965 - 0.258i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.758 + 1.83i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.258 - 0.965i)T^{2} \)
61 \( 1 + (0.0862 + 0.0983i)T + (-0.130 + 0.991i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.793 + 0.608i)T^{2} \)
73 \( 1 + (0.491 + 0.735i)T + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.867 + 1.75i)T + (-0.608 - 0.793i)T^{2} \)
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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

−10.17910801431851050286733416977, −9.548006561262808888953958644523, −8.704965849615231372237808324286, −7.84476752020518687535171804238, −6.84064060959338266322773577874, −6.14334096197841808659410498199, −5.09060708654907516174771045286, −4.61965033639562709928060585689, −3.25787014157107241028677383015, −1.82149684593980371713837120766, 1.83784293107116822463340349460, 2.46685478872041959786092350438, 3.87105022477490131027804923835, 4.75608846062987525546423600955, 5.78395686169762846537149379763, 6.64433598444002397339320099854, 7.34189716278800901328133808556, 9.117468946206165966471020112152, 9.483920827885940228686656499076, 10.45355708284576724435363012923

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