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Properties

Label 2-884-884.167-c0-0-0
Degree $2$
Conductor $884$
Sign $0.525 + 0.850i$
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 + (0.867 + 0.172i)5-s + (0.382 + 0.923i)8-s + (0.793 − 0.608i)9-s + (0.0578 − 0.882i)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 + (−0.882 + 0.0578i)20-s + (−0.200 − 0.0832i)25-s + (0.965 − 0.258i)26-s + (0.882 − 1.78i)29-s + (−0.608 − 0.793i)32-s + ⋯
L(s)  = 1  + (−0.130 − 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.867 + 0.172i)5-s + (0.382 + 0.923i)8-s + (0.793 − 0.608i)9-s + (0.0578 − 0.882i)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 + (−0.882 + 0.0578i)20-s + (−0.200 − 0.0832i)25-s + (0.965 − 0.258i)26-s + (0.882 − 1.78i)29-s + (−0.608 − 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024925364\)
\(L(\frac12)\) \(\approx\) \(1.024925364\)
\(L(1)\) 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 \( 1 + (-0.867 - 0.172i)T + (0.923 + 0.382i)T^{2} \)
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 \( 1 + (-0.882 + 1.78i)T + (-0.608 - 0.793i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.837 + 1.69i)T + (-0.608 - 0.793i)T^{2} \)
41 \( 1 + (0.991 - 0.869i)T + (0.130 - 0.991i)T^{2} \)
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 \( 1 + (-0.284 - 0.576i)T + (-0.608 + 0.793i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.991 - 0.130i)T^{2} \)
73 \( 1 + (0.293 - 1.47i)T + (-0.923 - 0.382i)T^{2} \)
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 \( 1 + (-0.835 - 0.732i)T + (0.130 + 0.991i)T^{2} \)
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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

−9.955865751317514387717516084965, −9.737586355742015497864350471911, −8.822844179091258821088669540220, −7.88745130506819672643659242949, −6.62802692665145796245464481140, −5.88216308296129665614596372515, −4.50988035226696311027276852245, −3.85253555220343489980719929677, −2.44007585398089797582045368665, −1.50229519833480092023992526069, 1.47525253806641032483278124680, 3.17324978313828579517364688418, 4.73432116217977603069057529119, 5.17891255116328946588071038880, 6.23708480972859863235687944431, 7.03024808995913135704711436973, 7.890468106656273722058470443910, 8.689592977178332024500509634428, 9.665350856746248342836262471936, 10.11659749215483961021631542030

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