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Properties

Label 2-896-16.5-c1-0-4
Degree $2$
Conductor $896$
Sign $0.255 - 0.966i$
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  + (0.416 − 0.416i)3-s + (1.13 + 1.13i)5-s − i·7-s + 2.65i·9-s + (3.85 + 3.85i)11-s + (−4.66 + 4.66i)13-s + 0.943·15-s − 5.33·17-s + (−2.55 + 2.55i)19-s + (−0.416 − 0.416i)21-s − 2.60i·23-s − 2.43i·25-s + (2.35 + 2.35i)27-s + (1.22 − 1.22i)29-s + 0.833·31-s + ⋯
L(s)  = 1  + (0.240 − 0.240i)3-s + (0.506 + 0.506i)5-s − 0.377i·7-s + 0.884i·9-s + (1.16 + 1.16i)11-s + (−1.29 + 1.29i)13-s + 0.243·15-s − 1.29·17-s + (−0.587 + 0.587i)19-s + (−0.0909 − 0.0909i)21-s − 0.543i·23-s − 0.487i·25-s + (0.453 + 0.453i)27-s + (0.227 − 0.227i)29-s + 0.149·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.255 - 0.966i$
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: \((2,\ 896,\ (\ :1/2),\ 0.255 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28820 + 0.991668i\)
\(L(\frac12)\) \(\approx\) \(1.28820 + 0.991668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.416 + 0.416i)T - 3iT^{2} \)
5 \( 1 + (-1.13 - 1.13i)T + 5iT^{2} \)
11 \( 1 + (-3.85 - 3.85i)T + 11iT^{2} \)
13 \( 1 + (4.66 - 4.66i)T - 13iT^{2} \)
17 \( 1 + 5.33T + 17T^{2} \)
19 \( 1 + (2.55 - 2.55i)T - 19iT^{2} \)
23 \( 1 + 2.60iT - 23T^{2} \)
29 \( 1 + (-1.22 + 1.22i)T - 29iT^{2} \)
31 \( 1 - 0.833T + 31T^{2} \)
37 \( 1 + (-4.42 - 4.42i)T + 37iT^{2} \)
41 \( 1 - 0.263iT - 41T^{2} \)
43 \( 1 + (-1.25 - 1.25i)T + 43iT^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + (0.0476 + 0.0476i)T + 53iT^{2} \)
59 \( 1 + (3.60 + 3.60i)T + 59iT^{2} \)
61 \( 1 + (4.46 - 4.46i)T - 61iT^{2} \)
67 \( 1 + (-9.50 + 9.50i)T - 67iT^{2} \)
71 \( 1 - 2.05iT - 71T^{2} \)
73 \( 1 - 5.48iT - 73T^{2} \)
79 \( 1 + 5.21T + 79T^{2} \)
83 \( 1 + (-5.84 + 5.84i)T - 83iT^{2} \)
89 \( 1 - 6.32iT - 89T^{2} \)
97 \( 1 - 18.8T + 97T^{2} \)
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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

−10.18212328112874211279823699009, −9.546062745411344106615067483181, −8.701448802260771601191450113262, −7.56582330827977766129877380306, −6.84627793312944451067582913899, −6.32358912180147996193168784127, −4.64735710739161615831021015709, −4.29264580313671627926319611670, −2.42624519023667660106079777858, −1.89938786586534582987666030740, 0.74247267911880027392747232761, 2.41905449677772470354695322056, 3.48839088264108625329444687649, 4.58438467265121121520652182402, 5.64224784336321305255308061696, 6.34220679105867416915929801758, 7.38106873648950965246197791997, 8.659648506942986094851657613431, 9.036356387776053172478223043752, 9.693437335167267905921329213758

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