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Properties

Label 2-882-63.5-c1-0-3
Degree $2$
Conductor $882$
Sign $0.202 - 0.979i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (−0.848 − 1.50i)3-s − 4-s + (1.96 + 3.39i)5-s + (−1.50 + 0.848i)6-s + i·8-s + (−1.55 + 2.56i)9-s + (3.39 − 1.96i)10-s + (−3.02 − 1.74i)11-s + (0.848 + 1.50i)12-s + (−2.18 − 1.26i)13-s + (3.46 − 5.84i)15-s + 16-s + (1.62 + 2.80i)17-s + (2.56 + 1.55i)18-s + (−1.85 − 1.07i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.490 − 0.871i)3-s − 0.5·4-s + (0.876 + 1.51i)5-s + (−0.616 + 0.346i)6-s + 0.353i·8-s + (−0.519 + 0.854i)9-s + (1.07 − 0.620i)10-s + (−0.912 − 0.527i)11-s + (0.245 + 0.435i)12-s + (−0.607 − 0.350i)13-s + (0.894 − 1.50i)15-s + 0.250·16-s + (0.392 + 0.680i)17-s + (0.604 + 0.367i)18-s + (−0.425 − 0.245i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.202 - 0.979i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443145 + 0.361046i\)
\(L(\frac12)\) \(\approx\) \(0.443145 + 0.361046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.848 + 1.50i)T \)
7 \( 1 \)
good5 \( 1 + (-1.96 - 3.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.02 + 1.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.18 + 1.26i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.62 - 2.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.85 + 1.07i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.15 - 4.70i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.38 - 4.26i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.29iT - 31T^{2} \)
37 \( 1 + (1.31 - 2.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.541 - 0.937i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.98 + 6.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.47T + 47T^{2} \)
53 \( 1 + (-9.31 + 5.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.34T + 59T^{2} \)
61 \( 1 - 4.15iT - 61T^{2} \)
67 \( 1 - 0.712T + 67T^{2} \)
71 \( 1 - 4.96iT - 71T^{2} \)
73 \( 1 + (5.69 - 3.28i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 8.67T + 79T^{2} \)
83 \( 1 + (-0.694 - 1.20i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.96 - 13.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.18 - 1.83i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.34024017196674641519117416399, −10.02775490399229474238679585676, −8.605040054903181052229822547782, −7.61675688572099595428300858248, −6.94160046293116666186592994294, −5.78360874225566705330749537895, −5.47394887272179579823396517125, −3.59071628780178793699121140685, −2.57379365257157840335613889671, −1.82130396655572865974813303393, 0.27015982058479444103281303995, 2.19861903576181074582032529930, 4.17029016042314220852887844112, 4.73132750128163028379069630771, 5.58578941931629536939011820698, 6.05610053458470822446397234966, 7.47862245390843426541993534026, 8.405592977849002715910940140244, 9.217952153985897689217750224101, 9.839061845412322393711165298341

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