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Properties

Label 2-884-884.227-c0-0-0
Degree $2$
Conductor $884$
Sign $-0.995 - 0.0938i$
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  + (0.130 − 0.991i)2-s + (−0.965 − 0.258i)4-s + (−0.349 − 1.75i)5-s + (−0.382 + 0.923i)8-s + (−0.793 − 0.608i)9-s + (−1.78 + 0.117i)10-s + (−0.130 + 0.991i)13-s + (0.866 + 0.5i)16-s + (−0.258 − 0.965i)17-s + (−0.707 + 0.707i)18-s + (−0.117 + 1.78i)20-s + (−2.04 + 0.848i)25-s + (0.965 + 0.258i)26-s + (0.117 − 0.0578i)29-s + (0.608 − 0.793i)32-s + ⋯
L(s)  = 1  + (0.130 − 0.991i)2-s + (−0.965 − 0.258i)4-s + (−0.349 − 1.75i)5-s + (−0.382 + 0.923i)8-s + (−0.793 − 0.608i)9-s + (−1.78 + 0.117i)10-s + (−0.130 + 0.991i)13-s + (0.866 + 0.5i)16-s + (−0.258 − 0.965i)17-s + (−0.707 + 0.707i)18-s + (−0.117 + 1.78i)20-s + (−2.04 + 0.848i)25-s + (0.965 + 0.258i)26-s + (0.117 − 0.0578i)29-s + (0.608 − 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $-0.995 - 0.0938i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ -0.995 - 0.0938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7045657093\)
\(L(\frac12)\) \(\approx\) \(0.7045657093\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 + 0.991i)T \)
13 \( 1 + (0.130 - 0.991i)T \)
17 \( 1 + (0.258 + 0.965i)T \)
good3 \( 1 + (0.793 + 0.608i)T^{2} \)
5 \( 1 + (0.349 + 1.75i)T + (-0.923 + 0.382i)T^{2} \)
7 \( 1 + (-0.793 + 0.608i)T^{2} \)
11 \( 1 + (0.991 - 0.130i)T^{2} \)
19 \( 1 + (-0.258 - 0.965i)T^{2} \)
23 \( 1 + (-0.130 - 0.991i)T^{2} \)
29 \( 1 + (-0.117 + 0.0578i)T + (0.608 - 0.793i)T^{2} \)
31 \( 1 + (-0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.576 + 0.284i)T + (0.608 - 0.793i)T^{2} \)
41 \( 1 + (-0.991 + 1.13i)T + (-0.130 - 0.991i)T^{2} \)
43 \( 1 + (0.258 + 0.965i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.46 + 0.607i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.965 + 0.258i)T^{2} \)
61 \( 1 + (1.69 + 0.837i)T + (0.608 + 0.793i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.991 - 0.130i)T^{2} \)
73 \( 1 + (-1.29 + 0.257i)T + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.923 + 0.382i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.09 - 1.25i)T + (-0.130 + 0.991i)T^{2} \)
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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

−9.613611491126927285638461110368, −9.161885537058223625414728380008, −8.655347553170495417063776520341, −7.67652317648602331425973239629, −6.14088080237894528064797550808, −5.12485158064625145226507120833, −4.49926981327737304357380440978, −3.54623509175289175792879659508, −2.10673081777600952742298916992, −0.67658122210769185361920370582, 2.68942125255138507769682205145, 3.47355857029573059996948176226, 4.67535488043471157322675693906, 5.98758139372823297077465064527, 6.29671169876614360268430722615, 7.61856504065039661049344008857, 7.76643081472926009406505922086, 8.853334897190696001517749706352, 10.04213271521591839308447693189, 10.68356438384657936976923291057

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