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Properties

Label 2-884-884.691-c0-0-0
Degree $2$
Conductor $884$
Sign $0.857 + 0.514i$
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  + (−0.793 − 0.608i)2-s + (0.258 + 0.965i)4-s + (−1.95 + 0.389i)5-s + (0.382 − 0.923i)8-s + (0.130 − 0.991i)9-s + (1.78 + 0.882i)10-s + (0.793 + 0.608i)13-s + (−0.866 + 0.499i)16-s + (0.965 + 0.258i)17-s + (−0.707 + 0.707i)18-s + (−0.882 − 1.78i)20-s + (2.75 − 1.14i)25-s + (−0.258 − 0.965i)26-s + (0.882 + 0.0578i)29-s + (0.991 + 0.130i)32-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)2-s + (0.258 + 0.965i)4-s + (−1.95 + 0.389i)5-s + (0.382 − 0.923i)8-s + (0.130 − 0.991i)9-s + (1.78 + 0.882i)10-s + (0.793 + 0.608i)13-s + (−0.866 + 0.499i)16-s + (0.965 + 0.258i)17-s + (−0.707 + 0.707i)18-s + (−0.882 − 1.78i)20-s + (2.75 − 1.14i)25-s + (−0.258 − 0.965i)26-s + (0.882 + 0.0578i)29-s + (0.991 + 0.130i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ 0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5105753335\)
\(L(\frac12)\) \(\approx\) \(0.5105753335\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + (-0.793 - 0.608i)T \)
17 \( 1 + (-0.965 - 0.258i)T \)
good3 \( 1 + (-0.130 + 0.991i)T^{2} \)
5 \( 1 + (1.95 - 0.389i)T + (0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.130 + 0.991i)T^{2} \)
11 \( 1 + (0.608 + 0.793i)T^{2} \)
19 \( 1 + (0.965 + 0.258i)T^{2} \)
23 \( 1 + (0.793 - 0.608i)T^{2} \)
29 \( 1 + (-0.882 - 0.0578i)T + (0.991 + 0.130i)T^{2} \)
31 \( 1 + (0.382 + 0.923i)T^{2} \)
37 \( 1 + (-1.50 - 0.0983i)T + (0.991 + 0.130i)T^{2} \)
41 \( 1 + (-0.608 + 0.206i)T + (0.793 - 0.608i)T^{2} \)
43 \( 1 + (-0.965 - 0.258i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.241 + 0.0999i)T + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.258 - 0.965i)T^{2} \)
61 \( 1 + (1.31 - 0.0862i)T + (0.991 - 0.130i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.608 + 0.793i)T^{2} \)
73 \( 1 + (-0.369 - 1.85i)T + (-0.923 + 0.382i)T^{2} \)
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 \( 1 + (1.05 + 0.357i)T + (0.793 + 0.608i)T^{2} \)
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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

−10.39185820752908827285247630158, −9.383986480901261918934048753230, −8.514534757419022262830166865894, −7.927298007446046561339862633672, −7.11341366780422025495092752682, −6.34610109180698555172639850286, −4.37154343603054024531774403726, −3.73371503552015890613273920740, −2.98185995064489374579029723067, −0.969399870188528542905061957698, 1.01720413761775141385137351185, 3.01728134921855214849962639074, 4.32996759377928015455564543040, 5.10792687689962175911652994640, 6.26640970115232261642099100109, 7.56584713013871240675406265375, 7.78007363078432282396051638162, 8.422005343437679564591130970861, 9.349049939650049044428293108490, 10.55470533260027947526825396671

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