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Properties

Label 2-896-896.405-c0-0-0
Degree $2$
Conductor $896$
Sign $-0.0490 - 0.998i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.555 + 0.831i)2-s + (−0.382 + 0.923i)4-s + (0.980 + 0.195i)7-s + (−0.980 + 0.195i)8-s + (0.195 + 0.980i)9-s + (0.0924 − 0.938i)11-s + (0.382 + 0.923i)14-s + (−0.707 − 0.707i)16-s + (−0.707 + 0.707i)18-s + (0.831 − 0.444i)22-s + (−0.216 + 0.324i)23-s + (−0.831 + 0.555i)25-s + (−0.555 + 0.831i)28-s + (−0.577 + 0.0569i)29-s + (0.195 − 0.980i)32-s + ⋯
L(s)  = 1  + (0.555 + 0.831i)2-s + (−0.382 + 0.923i)4-s + (0.980 + 0.195i)7-s + (−0.980 + 0.195i)8-s + (0.195 + 0.980i)9-s + (0.0924 − 0.938i)11-s + (0.382 + 0.923i)14-s + (−0.707 − 0.707i)16-s + (−0.707 + 0.707i)18-s + (0.831 − 0.444i)22-s + (−0.216 + 0.324i)23-s + (−0.831 + 0.555i)25-s + (−0.555 + 0.831i)28-s + (−0.577 + 0.0569i)29-s + (0.195 − 0.980i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0490 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0490 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.0490 - 0.998i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ -0.0490 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.341730069\)
\(L(\frac12)\) \(\approx\) \(1.341730069\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.555 - 0.831i)T \)
7 \( 1 + (-0.980 - 0.195i)T \)
good3 \( 1 + (-0.195 - 0.980i)T^{2} \)
5 \( 1 + (0.831 - 0.555i)T^{2} \)
11 \( 1 + (-0.0924 + 0.938i)T + (-0.980 - 0.195i)T^{2} \)
13 \( 1 + (-0.831 - 0.555i)T^{2} \)
17 \( 1 + (0.707 - 0.707i)T^{2} \)
19 \( 1 + (0.555 - 0.831i)T^{2} \)
23 \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (0.577 - 0.0569i)T + (0.980 - 0.195i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.172 + 0.0924i)T + (0.555 + 0.831i)T^{2} \)
41 \( 1 + (-0.382 - 0.923i)T^{2} \)
43 \( 1 + (-1.53 + 1.26i)T + (0.195 - 0.980i)T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T^{2} \)
53 \( 1 + (1.26 + 0.124i)T + (0.980 + 0.195i)T^{2} \)
59 \( 1 + (-0.831 + 0.555i)T^{2} \)
61 \( 1 + (0.195 + 0.980i)T^{2} \)
67 \( 1 + (-0.980 + 1.19i)T + (-0.195 - 0.980i)T^{2} \)
71 \( 1 + (-0.360 + 1.81i)T + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.923 + 0.382i)T^{2} \)
79 \( 1 + (1.53 + 0.636i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (-0.555 + 0.831i)T^{2} \)
89 \( 1 + (0.382 - 0.923i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.78234326998236613770467461763, −9.438898021450534440220069672406, −8.577579294451316054049413743815, −7.86195341812034556158442801378, −7.30347925003580063418004641239, −5.99589669545380873487174913413, −5.37802154903694270009417060447, −4.51081027263931723340698635248, −3.46720326072147309594054337955, −2.05441656444153624065452811060, 1.35333550966473036527470653453, 2.50362998171137964668882483268, 3.92985777897249610851822618478, 4.47020001260673813602774891192, 5.55136646691635955436585360773, 6.50479732017231085012631153244, 7.53841394261165561234128239948, 8.619030034816781705012428599714, 9.569752919284787545845361214561, 10.09910640595447816129381204608

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