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Properties

Label 2-896-8.5-c1-0-7
Degree $2$
Conductor $896$
Sign $-0.707 - 0.707i$
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  + 2i·3-s + 2i·5-s + 7-s − 9-s + 2i·13-s − 4·15-s + 6·17-s + 6i·19-s + 2i·21-s − 4·23-s + 25-s + 4i·27-s − 8·31-s + 2i·35-s − 8i·37-s + ⋯
L(s)  = 1  + 1.15i·3-s + 0.894i·5-s + 0.377·7-s − 0.333·9-s + 0.554i·13-s − 1.03·15-s + 1.45·17-s + 1.37i·19-s + 0.436i·21-s − 0.834·23-s + 0.200·25-s + 0.769i·27-s − 1.43·31-s + 0.338i·35-s − 1.31i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603762 + 1.45761i\)
\(L(\frac12)\) \(\approx\) \(0.603762 + 1.45761i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.40127408081591978929773660381, −9.804155830162207405517004001813, −8.956202068870515196987246586886, −7.86427692448794406308763970929, −7.11890937961443462564381487319, −5.92045022256968320597341900784, −5.15865914961395090343056253630, −3.91536554176197935018058155674, −3.44151730792668139078293444251, −1.88324751361067601798360210873, 0.797204427107746605432915597720, 1.77870064030860721788297359816, 3.17008152437631666866722841487, 4.62507158999568088575334581251, 5.42625603075705964719964372265, 6.42032517500321750045528698838, 7.42476312026384558923739073067, 7.982361374988612303386556882570, 8.770579645144117436471212270114, 9.719104836408782915988558686051

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