Properties

Label 2-896-896.59-c1-0-119
Degree $2$
Conductor $896$
Sign $-0.349 - 0.936i$
Analytic cond. $7.15459$
Root an. cond. $2.67480$
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  + (0.480 − 1.33i)2-s + (−0.125 − 0.134i)3-s + (−1.53 − 1.27i)4-s + (−0.493 − 1.08i)5-s + (−0.238 + 0.102i)6-s + (−0.369 + 2.61i)7-s + (−2.43 + 1.43i)8-s + (0.193 − 2.95i)9-s + (−1.68 + 0.132i)10-s + (−2.41 − 3.88i)11-s + (0.0216 + 0.367i)12-s + (0.274 + 2.78i)13-s + (3.30 + 1.75i)14-s + (−0.0841 + 0.203i)15-s + (0.729 + 3.93i)16-s + (−5.61 − 0.739i)17-s + ⋯
L(s)  = 1  + (0.339 − 0.940i)2-s + (−0.0725 − 0.0774i)3-s + (−0.768 − 0.639i)4-s + (−0.220 − 0.486i)5-s + (−0.0975 + 0.0419i)6-s + (−0.139 + 0.990i)7-s + (−0.862 + 0.505i)8-s + (0.0646 − 0.986i)9-s + (−0.532 + 0.0419i)10-s + (−0.727 − 1.17i)11-s + (0.00626 + 0.105i)12-s + (0.0761 + 0.773i)13-s + (0.883 + 0.467i)14-s + (−0.0217 + 0.0524i)15-s + (0.182 + 0.983i)16-s + (−1.36 − 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.349 - 0.936i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.154001 + 0.221801i\)
\(L(\frac12)\) \(\approx\) \(0.154001 + 0.221801i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.480 + 1.33i)T \)
7 \( 1 + (0.369 - 2.61i)T \)
good3 \( 1 + (0.125 + 0.134i)T + (-0.196 + 2.99i)T^{2} \)
5 \( 1 + (0.493 + 1.08i)T + (-3.29 + 3.75i)T^{2} \)
11 \( 1 + (2.41 + 3.88i)T + (-4.86 + 9.86i)T^{2} \)
13 \( 1 + (-0.274 - 2.78i)T + (-12.7 + 2.53i)T^{2} \)
17 \( 1 + (5.61 + 0.739i)T + (16.4 + 4.39i)T^{2} \)
19 \( 1 + (0.846 - 5.12i)T + (-17.9 - 6.10i)T^{2} \)
23 \( 1 + (-0.0430 - 0.0490i)T + (-3.00 + 22.8i)T^{2} \)
29 \( 1 + (0.744 + 1.39i)T + (-16.1 + 24.1i)T^{2} \)
31 \( 1 + (6.49 - 1.74i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.69 - 7.15i)T + (-27.8 - 24.3i)T^{2} \)
41 \( 1 + (0.874 + 4.39i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (4.44 - 1.34i)T + (35.7 - 23.8i)T^{2} \)
47 \( 1 + (3.64 - 4.74i)T + (-12.1 - 45.3i)T^{2} \)
53 \( 1 + (2.59 + 4.18i)T + (-23.4 + 47.5i)T^{2} \)
59 \( 1 + (-2.19 + 3.06i)T + (-18.9 - 55.8i)T^{2} \)
61 \( 1 + (-11.4 + 2.67i)T + (54.7 - 26.9i)T^{2} \)
67 \( 1 + (-8.14 - 8.69i)T + (-4.38 + 66.8i)T^{2} \)
71 \( 1 + (0.169 + 0.253i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (12.4 - 6.13i)T + (44.4 - 57.9i)T^{2} \)
79 \( 1 + (2.04 + 15.5i)T + (-76.3 + 20.4i)T^{2} \)
83 \( 1 + (-1.81 + 2.21i)T + (-16.1 - 81.4i)T^{2} \)
89 \( 1 + (3.68 + 10.8i)T + (-70.6 + 54.1i)T^{2} \)
97 \( 1 + (-11.7 + 11.7i)T - 97iT^{2} \)
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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

−9.537148434590810489095659563754, −8.704954658907511678400418198680, −8.428059125687712442186940311734, −6.62866311049037969791704688530, −5.88500386252072493694860677385, −5.00018023095094882189248377597, −3.92186867937105646683548971080, −2.99145408656509897192019433151, −1.76245293026447873573480765719, −0.11070918920284811960016859185, 2.42227170433712181154505733384, 3.72647442732945434879404603829, 4.70205080011777640643543546857, 5.29585180880974467027432020578, 6.72995873101856984745850710464, 7.21175964309650490744320330083, 7.82728787032843262910840831571, 8.829962494650537900604516349080, 9.879315094964311890769663627979, 10.70702065690318449261111269110

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