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Properties

Label 2-884-884.583-c0-0-0
Degree $2$
Conductor $884$
Sign $0.796 + 0.604i$
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 + (0.534 − 0.357i)5-s + (−0.923 + 0.382i)8-s + (−0.608 − 0.793i)9-s + (−0.483 + 0.423i)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 + (0.423 − 0.483i)20-s + (−0.224 + 0.542i)25-s + (−0.965 + 0.258i)26-s + (−0.423 − 1.24i)29-s + (−0.793 + 0.608i)32-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)2-s + (0.965 − 0.258i)4-s + (0.534 − 0.357i)5-s + (−0.923 + 0.382i)8-s + (−0.608 − 0.793i)9-s + (−0.483 + 0.423i)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 + (0.423 − 0.483i)20-s + (−0.224 + 0.542i)25-s + (−0.965 + 0.258i)26-s + (−0.423 − 1.24i)29-s + (−0.793 + 0.608i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ 0.796 + 0.604i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7007627887\)
\(L(\frac12)\) \(\approx\) \(0.7007627887\)
\(L(1)\) 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 \( 1 + (-0.534 + 0.357i)T + (0.382 - 0.923i)T^{2} \)
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 \( 1 + (0.423 + 1.24i)T + (-0.793 + 0.608i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (-0.284 - 0.837i)T + (-0.793 + 0.608i)T^{2} \)
41 \( 1 + (-0.130 - 0.00855i)T + (0.991 + 0.130i)T^{2} \)
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 \( 1 + (-0.576 + 1.69i)T + (-0.793 - 0.608i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.130 - 0.991i)T^{2} \)
73 \( 1 + (-1.10 - 1.65i)T + (-0.382 + 0.923i)T^{2} \)
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 \( 1 + (1.95 - 0.128i)T + (0.991 - 0.130i)T^{2} \)
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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

−9.914740543668325566922362590552, −9.430620934491540746641486829286, −8.680551878970562724120460405972, −7.929236565036440271570893120066, −6.85528079450983969173457777101, −6.03291829093666654352236930272, −5.34770757647985918951984705038, −3.66821470531012275759444837315, −2.50242903189510211828558324598, −1.05449892289509338374692787430, 1.63117570151215947033392601215, 2.67484247857198001475538168155, 3.84375419378168967002804885474, 5.53528355048972463032358527376, 6.18069640523327552585241566736, 7.14790105459554208571004348078, 8.127776157323945169893572058558, 8.699222395642828823758117375467, 9.577498160498794327721513767340, 10.67224489319853172663383746293

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