Properties

Label 2-3479-497.354-c0-0-8
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  + (0.900 + 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

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

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: $\chi_{3479} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3479,\ (\ :0),\ -0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.863415741\)
\(L(\frac12)\) \(\approx\) \(1.863415741\)
\(L(1)\) 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

−8.441034267472186927665451690212, −8.078942307271658332946867142315, −7.56559821927742076449693893252, −6.89800560543152964627633158942, −5.89635184057211227287610790269, −5.35443709383948362727422562914, −4.51973035955392498793463074680, −4.07739474649652123509236687199, −3.01985465048293395942077836386, −1.42180460640654870876114204339, 0.897539491788389904159860570740, 2.37721643520263157803459343976, 2.99641673417018749870916723505, 3.69986542689964664009349716993, 4.25067760423411041997258828861, 5.10853056465199984290097406416, 6.04465302146776748786762552219, 6.98580662566909687561380056011, 7.61973360320699801492304775193, 9.069431769878939429752676025592

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