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Properties

Label 2-896-896.59-c1-0-106
Degree $2$
Conductor $896$
Sign $-0.982 - 0.183i$
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  + (−1.30 + 0.538i)2-s + (−0.653 − 0.697i)3-s + (1.41 − 1.40i)4-s + (−0.673 − 1.48i)5-s + (1.22 + 0.559i)6-s + (−1.12 − 2.39i)7-s + (−1.09 + 2.60i)8-s + (0.136 − 2.08i)9-s + (1.68 + 1.58i)10-s + (−1.17 − 1.88i)11-s + (−1.90 − 0.0693i)12-s + (−0.0286 − 0.291i)13-s + (2.75 + 2.53i)14-s + (−0.596 + 1.44i)15-s + (0.0287 − 3.99i)16-s + (−1.92 − 0.253i)17-s + ⋯
L(s)  = 1  + (−0.924 + 0.381i)2-s + (−0.377 − 0.402i)3-s + (0.709 − 0.704i)4-s + (−0.301 − 0.664i)5-s + (0.501 + 0.228i)6-s + (−0.423 − 0.905i)7-s + (−0.387 + 0.921i)8-s + (0.0455 − 0.694i)9-s + (0.531 + 0.499i)10-s + (−0.352 − 0.567i)11-s + (−0.551 − 0.0200i)12-s + (−0.00795 − 0.0807i)13-s + (0.736 + 0.676i)14-s + (−0.154 + 0.371i)15-s + (0.00718 − 0.999i)16-s + (−0.467 − 0.0615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.982 - 0.183i$
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: \((2,\ 896,\ (\ :1/2),\ -0.982 - 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0343586 + 0.370525i\)
\(L(\frac12)\) \(\approx\) \(0.0343586 + 0.370525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.538i)T \)
7 \( 1 + (1.12 + 2.39i)T \)
good3 \( 1 + (0.653 + 0.697i)T + (-0.196 + 2.99i)T^{2} \)
5 \( 1 + (0.673 + 1.48i)T + (-3.29 + 3.75i)T^{2} \)
11 \( 1 + (1.17 + 1.88i)T + (-4.86 + 9.86i)T^{2} \)
13 \( 1 + (0.0286 + 0.291i)T + (-12.7 + 2.53i)T^{2} \)
17 \( 1 + (1.92 + 0.253i)T + (16.4 + 4.39i)T^{2} \)
19 \( 1 + (-0.252 + 1.52i)T + (-17.9 - 6.10i)T^{2} \)
23 \( 1 + (1.00 + 1.14i)T + (-3.00 + 22.8i)T^{2} \)
29 \( 1 + (-2.84 - 5.32i)T + (-16.1 + 24.1i)T^{2} \)
31 \( 1 + (-0.568 + 0.152i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.272 + 0.724i)T + (-27.8 - 24.3i)T^{2} \)
41 \( 1 + (-0.326 - 1.64i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (2.02 - 0.614i)T + (35.7 - 23.8i)T^{2} \)
47 \( 1 + (0.854 - 1.11i)T + (-12.1 - 45.3i)T^{2} \)
53 \( 1 + (-3.67 - 5.91i)T + (-23.4 + 47.5i)T^{2} \)
59 \( 1 + (1.12 - 1.57i)T + (-18.9 - 55.8i)T^{2} \)
61 \( 1 + (-7.78 + 1.81i)T + (54.7 - 26.9i)T^{2} \)
67 \( 1 + (3.50 + 3.74i)T + (-4.38 + 66.8i)T^{2} \)
71 \( 1 + (6.37 + 9.54i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (1.97 - 0.975i)T + (44.4 - 57.9i)T^{2} \)
79 \( 1 + (-0.446 - 3.38i)T + (-76.3 + 20.4i)T^{2} \)
83 \( 1 + (11.1 - 13.5i)T + (-16.1 - 81.4i)T^{2} \)
89 \( 1 + (1.69 + 4.97i)T + (-70.6 + 54.1i)T^{2} \)
97 \( 1 + (5.68 - 5.68i)T - 97iT^{2} \)
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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

−9.549820067601021394299569105175, −8.780733859736285932466393811591, −8.034574942946671517334199058244, −7.05298028668759407778319540091, −6.52616166186109200769701795521, −5.54631567723364654501544989743, −4.39077768591045641863350867641, −2.99931883467550401289754933951, −1.20784883266320324896607392626, −0.27509886764973263405351429029, 2.04732733566319476021893463651, 2.91511899292319933018459225471, 4.14016244243202222287760126364, 5.41126362230972518221874065401, 6.44123539229074605327079529150, 7.30465386715491419494498428880, 8.139771699223673464177704415698, 8.975046676277958679969416957380, 10.02090133986012414214776765667, 10.28615527582386847292945004106

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