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Properties

Label 2-896-112.59-c1-0-13
Degree $2$
Conductor $896$
Sign $0.254 - 0.967i$
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  + (0.665 + 2.48i)3-s + (3.12 + 0.837i)5-s + (1.56 − 2.13i)7-s + (−3.13 + 1.80i)9-s + (−0.376 − 1.40i)11-s + (3.11 + 3.11i)13-s + 8.31i·15-s + (2.02 + 1.16i)17-s + (−4.40 − 1.18i)19-s + (6.34 + 2.45i)21-s + (−1.15 − 1.99i)23-s + (4.72 + 2.73i)25-s + (−1.12 − 1.12i)27-s + (1.55 − 1.55i)29-s + (3.88 − 6.73i)31-s + ⋯
L(s)  = 1  + (0.384 + 1.43i)3-s + (1.39 + 0.374i)5-s + (0.590 − 0.807i)7-s + (−1.04 + 0.602i)9-s + (−0.113 − 0.423i)11-s + (0.863 + 0.863i)13-s + 2.14i·15-s + (0.490 + 0.283i)17-s + (−1.01 − 0.270i)19-s + (1.38 + 0.536i)21-s + (−0.240 − 0.416i)23-s + (0.945 + 0.546i)25-s + (−0.216 − 0.216i)27-s + (0.288 − 0.288i)29-s + (0.698 − 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93022 + 1.48792i\)
\(L(\frac12)\) \(\approx\) \(1.93022 + 1.48792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-1.56 + 2.13i)T \)
good3 \( 1 + (-0.665 - 2.48i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-3.12 - 0.837i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.376 + 1.40i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.11 - 3.11i)T + 13iT^{2} \)
17 \( 1 + (-2.02 - 1.16i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.40 + 1.18i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.15 + 1.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.55 + 1.55i)T - 29iT^{2} \)
31 \( 1 + (-3.88 + 6.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.272 - 1.01i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.77T + 41T^{2} \)
43 \( 1 + (7.12 - 7.12i)T - 43iT^{2} \)
47 \( 1 + (-1.42 - 2.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.0 - 2.97i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.77 - 1.01i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.72 + 13.9i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.59 - 0.695i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.48T + 71T^{2} \)
73 \( 1 + (5.65 - 9.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.706 - 0.408i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.65 - 2.65i)T - 83iT^{2} \)
89 \( 1 + (2.40 + 4.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.86iT - 97T^{2} \)
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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

−10.10962190930649952566065693617, −9.731556658052814913973882307496, −8.779913685648369520145632164465, −8.080242613203584658917600329908, −6.59384594032740080982176278356, −5.95575033984300721455714569087, −4.75886604417233468620281900156, −4.10965026993759089271639473659, −2.99420097089205680955232295517, −1.69875621465105670930442228556, 1.36921155490156868395789223696, 1.98738721143320056441204065208, 3.03834136439544000732907677200, 4.94018213075854830780891515092, 5.78958503732126911029152289165, 6.39183648503447904030159294383, 7.42776553965094796563937473731, 8.453632688630326089366078541095, 8.707858490340477137736072185646, 9.905324193011988406380933607718

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