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Properties

Label 2-884-884.539-c0-0-0
Degree $2$
Conductor $884$
Sign $-0.316 + 0.948i$
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 + (−1.57 + 1.05i)5-s + (0.382 + 0.923i)8-s + (−0.991 + 0.130i)9-s + (0.608 − 1.79i)10-s + (0.130 − 0.991i)13-s + (−0.866 − 0.499i)16-s + (−0.707 − 0.707i)17-s + (0.707 − 0.707i)18-s + (0.608 + 1.79i)20-s + (0.989 − 2.38i)25-s + (0.499 + 0.866i)26-s + (−1.13 − 0.991i)29-s + (0.991 − 0.130i)32-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + (−1.57 + 1.05i)5-s + (0.382 + 0.923i)8-s + (−0.991 + 0.130i)9-s + (0.608 − 1.79i)10-s + (0.130 − 0.991i)13-s + (−0.866 − 0.499i)16-s + (−0.707 − 0.707i)17-s + (0.707 − 0.707i)18-s + (0.608 + 1.79i)20-s + (0.989 − 2.38i)25-s + (0.499 + 0.866i)26-s + (−1.13 − 0.991i)29-s + (0.991 − 0.130i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $-0.316 + 0.948i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ -0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07651707256\)
\(L(\frac12)\) \(\approx\) \(0.07651707256\)
\(L(1)\) 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 \( 1 + (-0.130 + 0.991i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.991 - 0.130i)T^{2} \)
5 \( 1 + (1.57 - 1.05i)T + (0.382 - 0.923i)T^{2} \)
7 \( 1 + (0.991 + 0.130i)T^{2} \)
11 \( 1 + (0.793 + 0.608i)T^{2} \)
19 \( 1 + (0.965 - 0.258i)T^{2} \)
23 \( 1 + (0.608 - 0.793i)T^{2} \)
29 \( 1 + (1.13 + 0.991i)T + (0.130 + 0.991i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (1.18 - 1.34i)T + (-0.130 - 0.991i)T^{2} \)
41 \( 1 + (0.882 + 1.78i)T + (-0.608 + 0.793i)T^{2} \)
43 \( 1 + (0.965 - 0.258i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.465 - 1.12i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.258 + 0.965i)T^{2} \)
61 \( 1 + (0.665 - 0.583i)T + (0.130 - 0.991i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.793 + 0.608i)T^{2} \)
73 \( 1 + (0.0726 + 0.108i)T + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.172 - 0.349i)T + (-0.608 - 0.793i)T^{2} \)
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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

−10.23828648788706980061144365499, −8.959533663751278411978635034203, −8.263544187305408879116222857737, −7.59944498342536825673976604550, −6.93770041092768816863429556870, −5.99186982470966428541102488406, −4.91266731239312475753581716506, −3.55504493166386888935362029316, −2.55896967328168504622226507528, −0.095323343507969894942606698695, 1.68364917829225103476831952547, 3.34692283054790453283983465266, 4.04239636032853597440718425534, 5.05584573532824370573785415979, 6.59722079464281346543099312711, 7.53069906102544199284452212614, 8.400043711882001231008697811401, 8.766879777419833857496364256427, 9.485145064167332859054721019069, 10.93121846851322372217883559457

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