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Properties

Label 2-896-112.109-c1-0-18
Degree $2$
Conductor $896$
Sign $0.980 - 0.198i$
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  + (2.51 − 0.672i)3-s + (2.91 + 0.780i)5-s + (−1.41 + 2.23i)7-s + (3.25 − 1.87i)9-s + (0.838 + 3.12i)11-s + (−2.52 − 2.52i)13-s + 7.83·15-s + (0.201 − 0.348i)17-s + (−0.373 + 1.39i)19-s + (−2.06 + 6.56i)21-s + (7.89 − 4.55i)23-s + (3.54 + 2.04i)25-s + (1.39 − 1.39i)27-s + (−1.47 − 1.47i)29-s + (−2.12 + 3.68i)31-s + ⋯
L(s)  = 1  + (1.44 − 0.388i)3-s + (1.30 + 0.348i)5-s + (−0.536 + 0.843i)7-s + (1.08 − 0.626i)9-s + (0.252 + 0.943i)11-s + (−0.699 − 0.699i)13-s + 2.02·15-s + (0.0487 − 0.0844i)17-s + (−0.0855 + 0.319i)19-s + (−0.449 + 1.43i)21-s + (1.64 − 0.949i)23-s + (0.708 + 0.408i)25-s + (0.267 − 0.267i)27-s + (−0.273 − 0.273i)29-s + (−0.381 + 0.661i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.980 - 0.198i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.94536 + 0.295836i\)
\(L(\frac12)\) \(\approx\) \(2.94536 + 0.295836i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 2.23i)T \)
good3 \( 1 + (-2.51 + 0.672i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-2.91 - 0.780i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.838 - 3.12i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.52 + 2.52i)T + 13iT^{2} \)
17 \( 1 + (-0.201 + 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.373 - 1.39i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.89 + 4.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.47 + 1.47i)T + 29iT^{2} \)
31 \( 1 + (2.12 - 3.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.94 + 0.520i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.96iT - 41T^{2} \)
43 \( 1 + (0.997 - 0.997i)T - 43iT^{2} \)
47 \( 1 + (-2.09 - 3.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.488 + 1.82i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.636 + 2.37i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.685 + 2.55i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (11.6 - 3.12i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.451iT - 71T^{2} \)
73 \( 1 + (9.40 + 5.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.31 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.742 - 0.742i)T + 83iT^{2} \)
89 \( 1 + (-11.1 + 6.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−9.931612846988528131000755294574, −9.141729460782941525557499036316, −8.796512908403267941439744752575, −7.54669067749940971709676062161, −6.88507607090797749132269577733, −5.90592744357771104263537155049, −4.89284817799676026826024665873, −3.27658050387748455512939661286, −2.54495766583449712939187975000, −1.83930601649694296795814905975, 1.43816478440978253822298181570, 2.69088386488608841748986688830, 3.51224460466881001548678345168, 4.59011484293467530951595716741, 5.71445266307502359226894551496, 6.79137002690731932204695247147, 7.61233115376319071773368460599, 8.779550224545904794099233918914, 9.267691473449180235558250926305, 9.750553988362867388190252909319

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