Properties

Label 2-896-896.69-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 + (−1.36 − 0.728i)11-s + (0.923 − 0.382i)14-s + (0.707 + 0.707i)16-s + (0.707 − 0.707i)18-s + (0.980 − 1.19i)22-s + (−0.324 − 1.63i)23-s + (−0.980 − 0.195i)25-s + (0.195 + 0.980i)28-s + (−0.0924 − 0.172i)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 + (−1.36 − 0.728i)11-s + (0.923 − 0.382i)14-s + (0.707 + 0.707i)16-s + (0.707 − 0.707i)18-s + (0.980 − 1.19i)22-s + (−0.324 − 1.63i)23-s + (−0.980 − 0.195i)25-s + (0.195 + 0.980i)28-s + (−0.0924 − 0.172i)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: $\chi_{896} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ 0.427 + 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4371379363\)
\(L(\frac12)\) \(\approx\) \(0.4371379363\)
\(L(1)\) 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 \( 1 + (1.36 + 0.728i)T + (0.555 + 0.831i)T^{2} \)
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 \( 1 + (0.0924 + 0.172i)T + (-0.555 + 0.831i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-1.11 - 1.36i)T + (-0.195 + 0.980i)T^{2} \)
41 \( 1 + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (-0.273 - 0.902i)T + (-0.831 + 0.555i)T^{2} \)
47 \( 1 + (0.707 - 0.707i)T^{2} \)
53 \( 1 + (-0.902 + 1.68i)T + (-0.555 - 0.831i)T^{2} \)
59 \( 1 + (-0.980 - 0.195i)T^{2} \)
61 \( 1 + (-0.831 - 0.555i)T^{2} \)
67 \( 1 + (0.555 + 0.168i)T + (0.831 + 0.555i)T^{2} \)
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

−10.10397112503661538089813049744, −9.222222793876138940235595339063, −8.207624296630967467988758965754, −7.83388728059061853692833521210, −6.56329711689591331039020199463, −6.09499150419981572512658551783, −5.06325356737193829185885937116, −3.98466839131066052686796422158, −2.85731327921526230397213453086, −0.43712567007360193083461238717, 2.11760821690747758599519481532, 2.77909637376431326714344090739, 3.96196216786550395570674062040, 5.33239601196896075119524131563, 5.67683742242700816561408902097, 7.46128048255211379862778245258, 8.037694345844274758496289883445, 9.060751756139679194036383665139, 9.662941169418501451574267796059, 10.48961438167336879760818706418

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