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Properties

Label 2-3479-497.212-c0-0-27
Degree $2$
Conductor $3479$
Sign $-0.701 - 0.712i$
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  + (0.900 − 1.56i)2-s + (−0.222 − 0.385i)3-s + (−1.12 − 1.94i)4-s + (0.623 − 1.07i)5-s − 0.801·6-s − 2.24·8-s + (0.400 − 0.694i)9-s + (−1.12 − 1.94i)10-s + (−0.499 + 0.866i)12-s − 0.554·15-s + (−0.900 + 1.56i)16-s + (−0.722 − 1.25i)18-s + (−0.900 + 1.56i)19-s − 2.80·20-s + (0.500 + 0.866i)24-s + (−0.277 − 0.480i)25-s + ⋯
L(s)  = 1  + (0.900 − 1.56i)2-s + (−0.222 − 0.385i)3-s + (−1.12 − 1.94i)4-s + (0.623 − 1.07i)5-s − 0.801·6-s − 2.24·8-s + (0.400 − 0.694i)9-s + (−1.12 − 1.94i)10-s + (−0.499 + 0.866i)12-s − 0.554·15-s + (−0.900 + 1.56i)16-s + (−0.722 − 1.25i)18-s + (−0.900 + 1.56i)19-s − 2.80·20-s + (0.500 + 0.866i)24-s + (−0.277 − 0.480i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3479\)    =    \(7^{2} \cdot 71\)
Sign: $-0.701 - 0.712i$
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: \((2,\ 3479,\ (\ :0),\ -0.701 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.902201406\)
\(L(\frac12)\) \(\approx\) \(1.902201406\)
\(L(1)\) 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 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 0.445T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
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 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.80T + T^{2} \)
89 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.618882499864878662960639675343, −7.60804166237758289274799410931, −6.36291971187884891284521333178, −5.75518319920454848193257037793, −5.17558666137708122296646687645, −4.09958803401910367310204421325, −3.82550209665770489410397967094, −2.43399814268385564900794662257, −1.67694288617284494050679342443, −0.884571948817849579735741265292, 2.27431945835833468254751870976, 3.15029593883712429225745145638, 4.24809350178725615926188140225, 4.77376104869833063517286069343, 5.55630617121303605875653510385, 6.34247893231097049426848624617, 6.76287519470325715082219709406, 7.47688872921635745652975759043, 8.143030459370813896336487253118, 9.069299078874325015472244763770

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