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Properties

Label 2-896-896.685-c0-0-0
Degree $2$
Conductor $896$
Sign $0.903 + 0.427i$
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  + (−0.980 + 0.195i)2-s + (0.923 − 0.382i)4-s + (0.831 + 0.555i)7-s + (−0.831 + 0.555i)8-s + (−0.555 − 0.831i)9-s + (0.577 − 1.90i)11-s + (−0.923 − 0.382i)14-s + (0.707 − 0.707i)16-s + (0.707 + 0.707i)18-s + (−0.195 + 1.98i)22-s + (−1.08 − 0.216i)23-s + (0.195 + 0.980i)25-s + (0.980 + 0.195i)28-s + (1.47 − 0.448i)29-s + (−0.555 + 0.831i)32-s + ⋯
L(s)  = 1  + (−0.980 + 0.195i)2-s + (0.923 − 0.382i)4-s + (0.831 + 0.555i)7-s + (−0.831 + 0.555i)8-s + (−0.555 − 0.831i)9-s + (0.577 − 1.90i)11-s + (−0.923 − 0.382i)14-s + (0.707 − 0.707i)16-s + (0.707 + 0.707i)18-s + (−0.195 + 1.98i)22-s + (−1.08 − 0.216i)23-s + (0.195 + 0.980i)25-s + (0.980 + 0.195i)28-s + (1.47 − 0.448i)29-s + (−0.555 + 0.831i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.903 + 0.427i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ 0.903 + 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6978523514\)
\(L(\frac12)\) \(\approx\) \(0.6978523514\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.980 - 0.195i)T \)
7 \( 1 + (-0.831 - 0.555i)T \)
good3 \( 1 + (0.555 + 0.831i)T^{2} \)
5 \( 1 + (-0.195 - 0.980i)T^{2} \)
11 \( 1 + (-0.577 + 1.90i)T + (-0.831 - 0.555i)T^{2} \)
13 \( 1 + (0.195 - 0.980i)T^{2} \)
17 \( 1 + (-0.707 - 0.707i)T^{2} \)
19 \( 1 + (-0.980 - 0.195i)T^{2} \)
23 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
29 \( 1 + (-1.47 + 0.448i)T + (0.831 - 0.555i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.0569 - 0.577i)T + (-0.980 + 0.195i)T^{2} \)
41 \( 1 + (0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.902 + 1.68i)T + (-0.555 + 0.831i)T^{2} \)
47 \( 1 + (0.707 + 0.707i)T^{2} \)
53 \( 1 + (-1.68 - 0.512i)T + (0.831 + 0.555i)T^{2} \)
59 \( 1 + (0.195 + 0.980i)T^{2} \)
61 \( 1 + (-0.555 - 0.831i)T^{2} \)
67 \( 1 + (-0.831 - 0.444i)T + (0.555 + 0.831i)T^{2} \)
71 \( 1 + (0.425 - 0.636i)T + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (-0.149 - 0.360i)T + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (0.980 + 0.195i)T^{2} \)
89 \( 1 + (-0.923 + 0.382i)T^{2} \)
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

−10.21447024458927994343672933187, −9.135895195307424229118327673176, −8.554542405464162037797023769561, −8.160951228310254993712647373874, −6.83401013490854556919713358548, −6.02738631215242691339858494558, −5.41171355618386908578883569194, −3.68674637273595465041086824178, −2.59719565561349382622181878261, −1.06235904301495902466601847447, 1.59719120759213681589720411310, 2.48999041705503352304867473775, 4.11459427672052244550643508241, 4.98454304857285747195775485911, 6.40109625679992229575998062523, 7.22845173347030464321691076977, 7.965365630906974042733495291504, 8.577159219521256182759346641821, 9.742516766053554616544143623636, 10.24103743255152468981431781299

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