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Properties

Label 2-896-28.27-c1-0-30
Degree $2$
Conductor $896$
Sign $-0.845 + 0.534i$
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  + 0.874·3-s − 3.70i·5-s + (−1.41 − 2.23i)7-s − 2.23·9-s + 3.23i·11-s + 0.874i·13-s − 3.23i·15-s − 4.57i·17-s − 1.95·19-s + (−1.23 − 1.95i)21-s − 1.23i·23-s − 8.70·25-s − 4.57·27-s − 2·29-s + 10.2·31-s + ⋯
L(s)  = 1  + 0.504·3-s − 1.65i·5-s + (−0.534 − 0.845i)7-s − 0.745·9-s + 0.975i·11-s + 0.242i·13-s − 0.835i·15-s − 1.10i·17-s − 0.448·19-s + (−0.269 − 0.426i)21-s − 0.257i·23-s − 1.74·25-s − 0.880·27-s − 0.371·29-s + 1.83·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.845 + 0.534i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.845 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305400 - 1.05423i\)
\(L(\frac12)\) \(\approx\) \(0.305400 - 1.05423i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 2.23i)T \)
good3 \( 1 - 0.874T + 3T^{2} \)
5 \( 1 + 3.70iT - 5T^{2} \)
11 \( 1 - 3.23iT - 11T^{2} \)
13 \( 1 - 0.874iT - 13T^{2} \)
17 \( 1 + 4.57iT - 17T^{2} \)
19 \( 1 + 1.95T + 19T^{2} \)
23 \( 1 + 1.23iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 3.90iT - 41T^{2} \)
43 \( 1 - 1.70iT - 43T^{2} \)
47 \( 1 + 8.07T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 8.94iT - 61T^{2} \)
67 \( 1 - 9.70iT - 67T^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 + 14.1iT - 73T^{2} \)
79 \( 1 + 12.4iT - 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 9.82iT - 89T^{2} \)
97 \( 1 - 4.57iT - 97T^{2} \)
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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

−9.588093383301745468038758600711, −8.958429788299529436309075805730, −8.239686129346459957581591073653, −7.36861625374541787083698736802, −6.36029762837596688212028724828, −5.07064852660337654481152238513, −4.51888826610682980342043049708, −3.37962029258533855201715818439, −1.96340014655408535178794482379, −0.46179733147297661444400625252, 2.27936874957916430277837892613, 3.06948334576679811002074066353, 3.67090105735052193011594692492, 5.54132539927007580249599892726, 6.22143090807189607469728668920, 6.88024615624690102734192431615, 8.222482630005076160882850008678, 8.529765355557943673842848838002, 9.733926598788514163756679091552, 10.43580426302653206016540190492

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