Properties

Label 2-896-896.59-c1-0-17
Degree $2$
Conductor $896$
Sign $-0.816 - 0.577i$
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  + (−1.35 + 0.393i)2-s + (2.30 + 2.45i)3-s + (1.69 − 1.06i)4-s + (−0.422 − 0.932i)5-s + (−4.09 − 2.43i)6-s + (−1.83 − 1.90i)7-s + (−1.87 + 2.11i)8-s + (−0.545 + 8.31i)9-s + (0.940 + 1.10i)10-s + (2.50 + 4.02i)11-s + (6.51 + 1.69i)12-s + (0.241 + 2.45i)13-s + (3.24 + 1.86i)14-s + (1.31 − 3.18i)15-s + (1.71 − 3.61i)16-s + (−6.48 − 0.853i)17-s + ⋯
L(s)  = 1  + (−0.960 + 0.278i)2-s + (1.32 + 1.41i)3-s + (0.845 − 0.534i)4-s + (−0.188 − 0.416i)5-s + (−1.67 − 0.992i)6-s + (−0.694 − 0.719i)7-s + (−0.662 + 0.748i)8-s + (−0.181 + 2.77i)9-s + (0.297 + 0.347i)10-s + (0.755 + 1.21i)11-s + (1.88 + 0.488i)12-s + (0.0669 + 0.679i)13-s + (0.867 + 0.497i)14-s + (0.340 − 0.822i)15-s + (0.428 − 0.903i)16-s + (−1.57 − 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.816 - 0.577i$
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: \((2,\ 896,\ (\ :1/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382346 + 1.20225i\)
\(L(\frac12)\) \(\approx\) \(0.382346 + 1.20225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.393i)T \)
7 \( 1 + (1.83 + 1.90i)T \)
good3 \( 1 + (-2.30 - 2.45i)T + (-0.196 + 2.99i)T^{2} \)
5 \( 1 + (0.422 + 0.932i)T + (-3.29 + 3.75i)T^{2} \)
11 \( 1 + (-2.50 - 4.02i)T + (-4.86 + 9.86i)T^{2} \)
13 \( 1 + (-0.241 - 2.45i)T + (-12.7 + 2.53i)T^{2} \)
17 \( 1 + (6.48 + 0.853i)T + (16.4 + 4.39i)T^{2} \)
19 \( 1 + (0.175 - 1.06i)T + (-17.9 - 6.10i)T^{2} \)
23 \( 1 + (-4.54 - 5.18i)T + (-3.00 + 22.8i)T^{2} \)
29 \( 1 + (2.82 + 5.28i)T + (-16.1 + 24.1i)T^{2} \)
31 \( 1 + (-4.44 + 1.19i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.93 - 5.15i)T + (-27.8 - 24.3i)T^{2} \)
41 \( 1 + (-1.25 - 6.31i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (2.11 - 0.642i)T + (35.7 - 23.8i)T^{2} \)
47 \( 1 + (3.92 - 5.11i)T + (-12.1 - 45.3i)T^{2} \)
53 \( 1 + (1.62 + 2.60i)T + (-23.4 + 47.5i)T^{2} \)
59 \( 1 + (-2.20 + 3.08i)T + (-18.9 - 55.8i)T^{2} \)
61 \( 1 + (-1.29 + 0.301i)T + (54.7 - 26.9i)T^{2} \)
67 \( 1 + (5.62 + 6.00i)T + (-4.38 + 66.8i)T^{2} \)
71 \( 1 + (-6.16 - 9.23i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (5.13 - 2.53i)T + (44.4 - 57.9i)T^{2} \)
79 \( 1 + (0.795 + 6.04i)T + (-76.3 + 20.4i)T^{2} \)
83 \( 1 + (-3.28 + 3.99i)T + (-16.1 - 81.4i)T^{2} \)
89 \( 1 + (-4.56 - 13.4i)T + (-70.6 + 54.1i)T^{2} \)
97 \( 1 + (-2.54 + 2.54i)T - 97iT^{2} \)
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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

−9.872644664036632500779148391584, −9.625565643901483093853665567627, −8.992527042716484223297424506034, −8.227447215528464538758453047638, −7.30490081573220470243239068880, −6.50014083237904374291161033627, −4.76035306464773000301158784715, −4.24627348435437133323057764863, −3.04536334225286897641079877439, −1.85773813805648736212106058801, 0.67880845114323196635991981678, 2.09993177612247076636036904689, 2.98558110734207266891178094905, 3.52190259718440606176252697887, 6.07340299461718233600324570016, 6.74350287321356363141756039026, 7.23085207986247067082531840281, 8.488153440519971671276695090119, 8.759379359919315482405566942126, 9.233471963107950201830548587911

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