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Properties

Label 2-884-884.275-c0-0-0
Degree $2$
Conductor $884$
Sign $0.973 + 0.230i$
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 + (−0.128 − 0.0255i)5-s + (0.923 − 0.382i)8-s + (−0.793 + 0.608i)9-s + (−0.130 − 0.00855i)10-s + (0.608 − 0.793i)13-s + (0.866 − 0.5i)16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)18-s + (−0.130 + 0.00855i)20-s + (−0.908 − 0.376i)25-s + (0.499 − 0.866i)26-s + (−1.60 − 0.793i)29-s + (0.793 − 0.608i)32-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (−0.128 − 0.0255i)5-s + (0.923 − 0.382i)8-s + (−0.793 + 0.608i)9-s + (−0.130 − 0.00855i)10-s + (0.608 − 0.793i)13-s + (0.866 − 0.5i)16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)18-s + (−0.130 + 0.00855i)20-s + (−0.908 − 0.376i)25-s + (0.499 − 0.866i)26-s + (−1.60 − 0.793i)29-s + (0.793 − 0.608i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ 0.973 + 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.729025616\)
\(L(\frac12)\) \(\approx\) \(1.729025616\)
\(L(1)\) 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 \( 1 + (-0.608 + 0.793i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.793 - 0.608i)T^{2} \)
5 \( 1 + (0.128 + 0.0255i)T + (0.923 + 0.382i)T^{2} \)
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 \( 1 + (1.60 + 0.793i)T + (0.608 + 0.793i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (0.583 - 1.18i)T + (-0.608 - 0.793i)T^{2} \)
41 \( 1 + (1.42 - 1.24i)T + (0.130 - 0.991i)T^{2} \)
43 \( 1 + (0.258 - 0.965i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.241 - 0.0999i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (-1.34 + 0.665i)T + (0.608 - 0.793i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.991 - 0.130i)T^{2} \)
73 \( 1 + (0.125 - 0.630i)T + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.25 + 1.09i)T + (0.130 + 0.991i)T^{2} \)
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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

−10.47314876675569639017585368331, −9.769383132362104334743670541611, −8.232401478186502629769520056745, −7.910602230171318161684094870993, −6.60700630593671245028900484440, −5.75514649242143155932851489888, −5.13877775299468694636619606507, −3.87906894996066424559888790116, −3.08193317453911474257013742958, −1.79588994319590548235528916936, 1.88515118792304746813798763042, 3.29919276247902408915219916640, 3.89194749315106329829788094089, 5.23213674491316552156922978052, 5.85007818858689963968519961502, 6.84564843568035042784618463169, 7.56702223516539338507426857305, 8.666383548139655288965849297100, 9.470988049539126236301162519778, 10.65623485923117598908814144118

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