Properties

Label 2-896-896.517-c0-0-0
Degree $2$
Conductor $896$
Sign $0.427 + 0.903i$
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  + (0.195 − 0.980i)2-s + (−0.923 − 0.382i)4-s + (0.555 + 0.831i)7-s + (−0.555 + 0.831i)8-s + (0.831 + 0.555i)9-s + (0.598 − 1.11i)11-s + (0.923 − 0.382i)14-s + (0.707 + 0.707i)16-s + (0.707 − 0.707i)18-s + (−0.980 − 0.804i)22-s + (−0.324 − 1.63i)23-s + (0.980 + 0.195i)25-s + (−0.195 − 0.980i)28-s + (−1.75 + 0.938i)29-s + (0.831 − 0.555i)32-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)2-s + (−0.923 − 0.382i)4-s + (0.555 + 0.831i)7-s + (−0.555 + 0.831i)8-s + (0.831 + 0.555i)9-s + (0.598 − 1.11i)11-s + (0.923 − 0.382i)14-s + (0.707 + 0.707i)16-s + (0.707 − 0.707i)18-s + (−0.980 − 0.804i)22-s + (−0.324 − 1.63i)23-s + (0.980 + 0.195i)25-s + (−0.195 − 0.980i)28-s + (−1.75 + 0.938i)29-s + (0.831 − 0.555i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.427 + 0.903i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ 0.427 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.117482229\)
\(L(\frac12)\) \(\approx\) \(1.117482229\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.195 + 0.980i)T \)
7 \( 1 + (-0.555 - 0.831i)T \)
good3 \( 1 + (-0.831 - 0.555i)T^{2} \)
5 \( 1 + (-0.980 - 0.195i)T^{2} \)
11 \( 1 + (-0.598 + 1.11i)T + (-0.555 - 0.831i)T^{2} \)
13 \( 1 + (0.980 - 0.195i)T^{2} \)
17 \( 1 + (-0.707 + 0.707i)T^{2} \)
19 \( 1 + (0.195 + 0.980i)T^{2} \)
23 \( 1 + (0.324 + 1.63i)T + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (1.75 - 0.938i)T + (0.555 - 0.831i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.728 + 0.598i)T + (0.195 - 0.980i)T^{2} \)
41 \( 1 + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (1.68 - 0.512i)T + (0.831 - 0.555i)T^{2} \)
47 \( 1 + (0.707 - 0.707i)T^{2} \)
53 \( 1 + (-0.512 - 0.273i)T + (0.555 + 0.831i)T^{2} \)
59 \( 1 + (0.980 + 0.195i)T^{2} \)
61 \( 1 + (0.831 + 0.555i)T^{2} \)
67 \( 1 + (-0.555 + 1.83i)T + (-0.831 - 0.555i)T^{2} \)
71 \( 1 + (0.636 - 0.425i)T + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (0.382 + 0.923i)T^{2} \)
79 \( 1 + (0.750 - 1.81i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (-0.195 - 0.980i)T^{2} \)
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

−10.40236874488143752584347722947, −9.326854495596192874891519327873, −8.714104614896546331478529975297, −7.954654871475566169326766489187, −6.57967067287108050890386722995, −5.53218078785755375523299200915, −4.75334125186112925992808748180, −3.73812044465765558586863221424, −2.56912683068754057524288460001, −1.45448663612628308723069310551, 1.50682452866242756158153507527, 3.65870363918694437442290120750, 4.28138766898870443097603704225, 5.16992068272891935662794895069, 6.34451969150137062668504555226, 7.24204844253952868329906609338, 7.52897176906051554964727834006, 8.693078466816705710100639137481, 9.694161032125574604539887425821, 10.02832865065181842903218598726

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