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Properties

Label 2-884-884.435-c0-0-0
Degree $2$
Conductor $884$
Sign $0.514 - 0.857i$
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  + (0.991 + 0.130i)2-s + (0.965 + 0.258i)4-s + (−1.05 + 1.57i)5-s + (0.923 + 0.382i)8-s + (0.608 − 0.793i)9-s + (−1.24 + 1.42i)10-s + (−0.991 − 0.130i)13-s + (0.866 + 0.5i)16-s + (0.258 + 0.965i)17-s + (0.707 − 0.707i)18-s + (−1.42 + 1.24i)20-s + (−0.989 − 2.38i)25-s + (−0.965 − 0.258i)26-s + (1.42 + 0.483i)29-s + (0.793 + 0.608i)32-s + ⋯
L(s)  = 1  + (0.991 + 0.130i)2-s + (0.965 + 0.258i)4-s + (−1.05 + 1.57i)5-s + (0.923 + 0.382i)8-s + (0.608 − 0.793i)9-s + (−1.24 + 1.42i)10-s + (−0.991 − 0.130i)13-s + (0.866 + 0.5i)16-s + (0.258 + 0.965i)17-s + (0.707 − 0.707i)18-s + (−1.42 + 1.24i)20-s + (−0.989 − 2.38i)25-s + (−0.965 − 0.258i)26-s + (1.42 + 0.483i)29-s + (0.793 + 0.608i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $0.514 - 0.857i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ 0.514 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.573204288\)
\(L(\frac12)\) \(\approx\) \(1.573204288\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.991 - 0.130i)T \)
13 \( 1 + (0.991 + 0.130i)T \)
17 \( 1 + (-0.258 - 0.965i)T \)
good3 \( 1 + (-0.608 + 0.793i)T^{2} \)
5 \( 1 + (1.05 - 1.57i)T + (-0.382 - 0.923i)T^{2} \)
7 \( 1 + (0.608 + 0.793i)T^{2} \)
11 \( 1 + (-0.130 - 0.991i)T^{2} \)
19 \( 1 + (0.258 + 0.965i)T^{2} \)
23 \( 1 + (-0.991 + 0.130i)T^{2} \)
29 \( 1 + (-1.42 - 0.483i)T + (0.793 + 0.608i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (1.69 + 0.576i)T + (0.793 + 0.608i)T^{2} \)
41 \( 1 + (0.130 + 1.99i)T + (-0.991 + 0.130i)T^{2} \)
43 \( 1 + (-0.258 - 0.965i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.465 + 1.12i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.965 - 0.258i)T^{2} \)
61 \( 1 + (-0.837 + 0.284i)T + (0.793 - 0.608i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.130 - 0.991i)T^{2} \)
73 \( 1 + (0.108 + 0.0726i)T + (0.382 + 0.923i)T^{2} \)
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 \( 1 + (-0.0255 + 0.389i)T + (-0.991 - 0.130i)T^{2} \)
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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

−10.43136410308595134864817397345, −10.21620842179705989837655430025, −8.511096951104201888301722838803, −7.52082462662928728125659967198, −6.93952197010888981849513948500, −6.40478754198276941645592615728, −5.10913071464820642499669646398, −3.87026581694118101851237901507, −3.46157191307068340073183549242, −2.28638156049252468253142693399, 1.36547704659359954381515137077, 2.87930976238846689943392801825, 4.24860594314681614438836273256, 4.76739774119123729931420085124, 5.32604444751674324418973665241, 6.82541182188118475574107126552, 7.65485975592592260891866960242, 8.256273381159598283669230979454, 9.464730925185316596471468564649, 10.27541067933828528939947758759

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