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Properties

Label 2-896-224.83-c1-0-29
Degree $2$
Conductor $896$
Sign $-0.999 + 0.0407i$
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  + (1.25 − 3.03i)3-s + (−0.824 − 1.98i)5-s + (0.0837 − 2.64i)7-s + (−5.52 − 5.52i)9-s + (0.623 − 0.258i)11-s + (−1.24 + 3.00i)13-s − 7.08·15-s + 2.68·17-s + (4.54 + 1.88i)19-s + (−7.92 − 3.58i)21-s + (0.871 − 0.871i)23-s + (0.256 − 0.256i)25-s + (−14.6 + 6.06i)27-s + (−2.60 + 6.29i)29-s + 4.75·31-s + ⋯
L(s)  = 1  + (0.726 − 1.75i)3-s + (−0.368 − 0.889i)5-s + (0.0316 − 0.999i)7-s + (−1.84 − 1.84i)9-s + (0.187 − 0.0778i)11-s + (−0.344 + 0.832i)13-s − 1.82·15-s + 0.650·17-s + (1.04 + 0.431i)19-s + (−1.73 − 0.781i)21-s + (0.181 − 0.181i)23-s + (0.0513 − 0.0513i)25-s + (−2.81 + 1.16i)27-s + (−0.483 + 1.16i)29-s + 0.853·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.999 + 0.0407i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.999 + 0.0407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0351782 - 1.72512i\)
\(L(\frac12)\) \(\approx\) \(0.0351782 - 1.72512i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0837 + 2.64i)T \)
good3 \( 1 + (-1.25 + 3.03i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.824 + 1.98i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (-0.623 + 0.258i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.24 - 3.00i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 2.68T + 17T^{2} \)
19 \( 1 + (-4.54 - 1.88i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.871 + 0.871i)T - 23iT^{2} \)
29 \( 1 + (2.60 - 6.29i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 4.75T + 31T^{2} \)
37 \( 1 + (5.97 - 2.47i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.37 + 3.37i)T - 41iT^{2} \)
43 \( 1 + (-11.8 + 4.92i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 3.01iT - 47T^{2} \)
53 \( 1 + (-1.62 - 3.92i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.08 - 0.863i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.36 - 2.22i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (6.52 + 2.70i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.16 + 9.16i)T + 71iT^{2} \)
73 \( 1 + (-3.68 + 3.68i)T - 73iT^{2} \)
79 \( 1 + 5.81T + 79T^{2} \)
83 \( 1 + (4.18 + 1.73i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-0.872 - 0.872i)T + 89iT^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−9.304278149438241094440818321180, −8.741660820899431478435468992444, −7.80623948004105603182689930412, −7.33568850910793623400891229319, −6.60889058823889535702207666237, −5.42794464571931604448054554615, −4.11192832649028131322013630001, −3.05761762830986786186581455460, −1.61568067808855747919650121843, −0.798617315533441978756994430879, 2.68003558278074296358196817064, 3.08439669521409364518419850117, 4.13338616940626576998728391656, 5.18752456460507389872549306935, 5.86758945002151462033760227304, 7.43245271099247924832852666038, 8.146967028002469093320820136052, 9.086189135009058738504426397708, 9.702548601342647365245976123007, 10.32536135965032690983782444883

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