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Properties

Label 2-896-8.5-c1-0-18
Degree $2$
Conductor $896$
Sign $i$
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  − 2.82i·5-s + 7-s + 3·9-s − 2.82i·11-s − 2.82i·13-s − 2·17-s + 5.65i·19-s − 8·23-s − 3.00·25-s − 5.65i·29-s + 8·31-s − 2.82i·35-s − 5.65i·37-s − 6·41-s − 2.82i·43-s + ⋯
L(s)  = 1  − 1.26i·5-s + 0.377·7-s + 9-s − 0.852i·11-s − 0.784i·13-s − 0.485·17-s + 1.29i·19-s − 1.66·23-s − 0.600·25-s − 1.05i·29-s + 1.43·31-s − 0.478i·35-s − 0.929i·37-s − 0.937·41-s − 0.431i·43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $i$
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: \((2,\ 896,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12043 - 1.12043i\)
\(L(\frac12)\) \(\approx\) \(1.12043 - 1.12043i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
5 \( 1 + 2.82iT - 5T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 5.65iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 5.65iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 2.82iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 - 2.82iT - 61T^{2} \)
67 \( 1 + 8.48iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 5.65iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.02802462690031950476795924093, −8.943860890634936168755032201344, −8.198751758066426565343521706796, −7.68733378258482915238904640559, −6.25879022327998988420816635302, −5.51390009111885104010347936693, −4.49654318322456261879393212640, −3.75681814440807246550724075687, −2.01743277983972991638362598809, −0.790629161122765033494096088883, 1.75808336217274324287862295603, 2.78266886524975051263788886418, 4.14438652358631518155955229946, 4.79900662613161598031710452961, 6.35318431597181664960310599789, 6.90943940678629335274499558614, 7.53356222130215830201456907092, 8.662483801228599624285453916280, 9.738954303920613400659949004480, 10.26001053216394522954091653823

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