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Properties

Label 2-896-16.5-c1-0-12
Degree $2$
Conductor $896$
Sign $0.577 + 0.816i$
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  + (−0.599 + 0.599i)3-s + (−0.974 − 0.974i)5-s − i·7-s + 2.28i·9-s + (−1.72 − 1.72i)11-s + (−1.90 + 1.90i)13-s + 1.16·15-s + 6.71·17-s + (2.94 − 2.94i)19-s + (0.599 + 0.599i)21-s − 5.29i·23-s − 3.09i·25-s + (−3.16 − 3.16i)27-s + (3.03 − 3.03i)29-s − 1.19·31-s + ⋯
L(s)  = 1  + (−0.346 + 0.346i)3-s + (−0.436 − 0.436i)5-s − 0.377i·7-s + 0.760i·9-s + (−0.519 − 0.519i)11-s + (−0.528 + 0.528i)13-s + 0.302·15-s + 1.62·17-s + (0.676 − 0.676i)19-s + (0.130 + 0.130i)21-s − 1.10i·23-s − 0.619i·25-s + (−0.609 − 0.609i)27-s + (0.562 − 0.562i)29-s − 0.215·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.964823 - 0.499409i\)
\(L(\frac12)\) \(\approx\) \(0.964823 - 0.499409i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + (0.599 - 0.599i)T - 3iT^{2} \)
5 \( 1 + (0.974 + 0.974i)T + 5iT^{2} \)
11 \( 1 + (1.72 + 1.72i)T + 11iT^{2} \)
13 \( 1 + (1.90 - 1.90i)T - 13iT^{2} \)
17 \( 1 - 6.71T + 17T^{2} \)
19 \( 1 + (-2.94 + 2.94i)T - 19iT^{2} \)
23 \( 1 + 5.29iT - 23T^{2} \)
29 \( 1 + (-3.03 + 3.03i)T - 29iT^{2} \)
31 \( 1 + 1.19T + 31T^{2} \)
37 \( 1 + (-2.25 - 2.25i)T + 37iT^{2} \)
41 \( 1 + 3.94iT - 41T^{2} \)
43 \( 1 + (7.02 + 7.02i)T + 43iT^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 + (-3.01 - 3.01i)T + 53iT^{2} \)
59 \( 1 + (-4.96 - 4.96i)T + 59iT^{2} \)
61 \( 1 + (-9.69 + 9.69i)T - 61iT^{2} \)
67 \( 1 + (-3.55 + 3.55i)T - 67iT^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 4.06T + 79T^{2} \)
83 \( 1 + (-9.17 + 9.17i)T - 83iT^{2} \)
89 \( 1 - 16.9iT - 89T^{2} \)
97 \( 1 + 2.51T + 97T^{2} \)
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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

−10.24287773051656067399468688851, −9.198107250602795543806612544880, −8.133390722165843989622380041474, −7.66425002108816944739468519545, −6.53937401701133922109298152641, −5.32841149504451399565454709361, −4.80574282420283093172953537958, −3.73673011054685062347473839573, −2.44566569933361301839225067267, −0.61985270458437933851390762624, 1.25548268707495406723962161966, 2.93214437523036000220754634405, 3.70074303968710541903907539507, 5.26050414774160510718678796809, 5.75581438119325539522978736709, 7.03605399674492698342900552190, 7.52904775424792047848006778163, 8.395312707515240892393581474287, 9.798276683476815230427061958481, 9.930824226505226046501809677453

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