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Properties

Label 2-896-16.5-c1-0-11
Degree $2$
Conductor $896$
Sign $-0.0497 + 0.998i$
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  + (−1.39 + 1.39i)3-s + (−2.16 − 2.16i)5-s + i·7-s − 0.871i·9-s + (3.09 + 3.09i)11-s + (−1.75 + 1.75i)13-s + 6.02·15-s − 5.20·17-s + (0.851 − 0.851i)19-s + (−1.39 − 1.39i)21-s − 6.15i·23-s + 4.37i·25-s + (−2.96 − 2.96i)27-s + (6.24 − 6.24i)29-s − 2.78·31-s + ⋯
L(s)  = 1  + (−0.803 + 0.803i)3-s + (−0.968 − 0.968i)5-s + 0.377i·7-s − 0.290i·9-s + (0.933 + 0.933i)11-s + (−0.486 + 0.486i)13-s + 1.55·15-s − 1.26·17-s + (0.195 − 0.195i)19-s + (−0.303 − 0.303i)21-s − 1.28i·23-s + 0.874i·25-s + (−0.570 − 0.570i)27-s + (1.15 − 1.15i)29-s − 0.499·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0497 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0497 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.0497 + 0.998i$
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: \((2,\ 896,\ (\ :1/2),\ -0.0497 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.271215 - 0.285067i\)
\(L(\frac12)\) \(\approx\) \(0.271215 - 0.285067i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 - 1.39i)T - 3iT^{2} \)
5 \( 1 + (2.16 + 2.16i)T + 5iT^{2} \)
11 \( 1 + (-3.09 - 3.09i)T + 11iT^{2} \)
13 \( 1 + (1.75 - 1.75i)T - 13iT^{2} \)
17 \( 1 + 5.20T + 17T^{2} \)
19 \( 1 + (-0.851 + 0.851i)T - 19iT^{2} \)
23 \( 1 + 6.15iT - 23T^{2} \)
29 \( 1 + (-6.24 + 6.24i)T - 29iT^{2} \)
31 \( 1 + 2.78T + 31T^{2} \)
37 \( 1 + (4.11 + 4.11i)T + 37iT^{2} \)
41 \( 1 + 6.32iT - 41T^{2} \)
43 \( 1 + (3.05 + 3.05i)T + 43iT^{2} \)
47 \( 1 - 3.60T + 47T^{2} \)
53 \( 1 + (5.28 + 5.28i)T + 53iT^{2} \)
59 \( 1 + (-7.13 - 7.13i)T + 59iT^{2} \)
61 \( 1 + (1.03 - 1.03i)T - 61iT^{2} \)
67 \( 1 + (-0.966 + 0.966i)T - 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 + 6.61T + 79T^{2} \)
83 \( 1 + (-7.41 + 7.41i)T - 83iT^{2} \)
89 \( 1 + 3.26iT - 89T^{2} \)
97 \( 1 + 7.66T + 97T^{2} \)
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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

−9.909910453586471617042848025910, −9.040164396152926282119901420353, −8.474653353106351853089025981780, −7.27490484762319948694759414006, −6.41321335035234714048030334858, −5.19076781308451974706128086875, −4.43236287213664526529624141277, −4.12020809295134428338153175538, −2.16405560363035262933432019503, −0.23062868086326687732534600083, 1.24562976280374328468699905114, 3.06616793240160524994813867013, 3.86219488901426598700627355544, 5.19135388252520544753821476085, 6.38428370184365808548445675552, 6.81068540299477719235379929627, 7.54154928516050040294081480033, 8.453454708826511554361824058544, 9.546843664952708192744489377102, 10.73159923385840966359655203929

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