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Properties

Label 2-882-9.4-c1-0-17
Degree $2$
Conductor $882$
Sign $-0.0315 + 0.999i$
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  + (−0.5 − 0.866i)2-s + (−0.707 − 1.58i)3-s + (−0.499 + 0.866i)4-s + (−1.72 + 2.98i)5-s + (−1.01 + 1.40i)6-s + 0.999·8-s + (−2.00 + 2.23i)9-s + 3.44·10-s + (−2 − 3.46i)11-s + (1.72 + 0.178i)12-s + (−2.12 + 3.67i)13-s + (5.93 + 0.614i)15-s + (−0.5 − 0.866i)16-s − 1.41·17-s + (2.93 + 0.614i)18-s + 6.27·19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.408 − 0.912i)3-s + (−0.249 + 0.433i)4-s + (−0.770 + 1.33i)5-s + (−0.414 + 0.572i)6-s + 0.353·8-s + (−0.666 + 0.745i)9-s + 1.08·10-s + (−0.603 − 1.04i)11-s + (0.497 + 0.0514i)12-s + (−0.588 + 1.01i)13-s + (1.53 + 0.158i)15-s + (−0.125 − 0.216i)16-s − 0.342·17-s + (0.692 + 0.144i)18-s + 1.43·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0315 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0315 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0315 + 0.999i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.0315 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502241 - 0.518366i\)
\(L(\frac12)\) \(\approx\) \(0.502241 - 0.518366i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.707 + 1.58i)T \)
7 \( 1 \)
good5 \( 1 + (1.72 - 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 - 3.67i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 6.27T + 19T^{2} \)
23 \( 1 + (-4.37 + 7.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.563 + 0.976i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.73 + 4.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.74T + 37T^{2} \)
41 \( 1 + (-2.82 + 4.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.563 + 0.976i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.03 + 3.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + (4.15 - 7.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.13 - 5.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.43 + 5.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.87T + 71T^{2} \)
73 \( 1 + 4.42T + 73T^{2} \)
79 \( 1 + (-0.936 - 1.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.32 + 2.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + (-7.50 - 13.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.27699174909698602122685059103, −9.018892291257453971190048669793, −8.105535506563281260031857954979, −7.32259012712251841335567166765, −6.79610513235896027160185162963, −5.71355121570505002106716890578, −4.36461293611791741386333111489, −3.02838704758719297496208954640, −2.42034476927706105155928745164, −0.56434663958296403578489086884, 0.937175470315551712895235315786, 3.19674746244636255027984640686, 4.51091266574658026852749863081, 5.05361191953342865474012648147, 5.64102460670974753899534275023, 7.20951841007316329763717289698, 7.82761514668162125097560738323, 8.717047081825804537160484227098, 9.572205996623057189242538106045, 9.973840305012260383332539092209

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