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Properties

Label 2-3479-497.212-c0-0-5
Degree $2$
Conductor $3479$
Sign $-0.605 + 0.795i$
Analytic cond. $1.73624$
Root an. cond. $1.31766$
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  + (−0.623 + 1.07i)2-s + (0.900 + 1.56i)3-s + (−0.277 − 0.480i)4-s + (0.222 − 0.385i)5-s − 2.24·6-s − 0.554·8-s + (−1.12 + 1.94i)9-s + (0.277 + 0.480i)10-s + (0.5 − 0.866i)12-s + 0.801·15-s + (0.623 − 1.07i)16-s + (−1.40 − 2.42i)18-s + (−0.623 + 1.07i)19-s − 0.246·20-s + (−0.499 − 0.866i)24-s + (0.400 + 0.694i)25-s + ⋯
L(s)  = 1  + (−0.623 + 1.07i)2-s + (0.900 + 1.56i)3-s + (−0.277 − 0.480i)4-s + (0.222 − 0.385i)5-s − 2.24·6-s − 0.554·8-s + (−1.12 + 1.94i)9-s + (0.277 + 0.480i)10-s + (0.5 − 0.866i)12-s + 0.801·15-s + (0.623 − 1.07i)16-s + (−1.40 − 2.42i)18-s + (−0.623 + 1.07i)19-s − 0.246·20-s + (−0.499 − 0.866i)24-s + (0.400 + 0.694i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3479\)    =    \(7^{2} \cdot 71\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(1.73624\)
Root analytic conductor: \(1.31766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3479} (1206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3479,\ (\ :0),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9987860816\)
\(L(\frac12)\) \(\approx\) \(0.9987860816\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( 1 - T \)
good2 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.80T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.24T + T^{2} \)
89 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.103741879172899535861141247797, −8.567069042567214700997870356116, −8.007953220972883558060137349087, −7.29564421987516657331491904895, −6.22464250103872967366285418216, −5.44753685293287407862999674700, −4.82706878622952990485850513606, −3.77128071265738050502252991390, −3.21980219694955352643043937979, −1.97742482905351543108478641403, 0.60028339318405292262407915233, 1.82308950240811652506616147193, 2.28724303925064485203233853192, 3.03948837158146282785114599234, 3.86896851509917259641837419447, 5.44701987169984167207563318262, 6.42060075824846008722289743709, 6.85646988087659797838544056487, 7.72056037792340291780917472820, 8.386905276077802420789197236599

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