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Properties

Label 2-3479-71.51-c0-0-0
Degree $2$
Conductor $3479$
Sign $0.368 + 0.929i$
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  + (0.400 + 1.75i)2-s + (−2.02 + 0.974i)4-s + (−1.40 − 1.75i)8-s + (−0.623 + 0.781i)9-s + (−0.678 − 0.541i)11-s + (1.12 − 1.40i)16-s + (−1.62 − 0.781i)18-s + (0.678 − 1.40i)22-s + (−0.846 − 0.193i)23-s − 25-s + (−1.62 + 0.781i)29-s + (0.900 + 0.433i)32-s + (0.5 − 2.19i)36-s + (−0.400 − 1.75i)37-s + (0.0990 + 0.433i)43-s + (1.90 + 0.433i)44-s + ⋯
L(s)  = 1  + (0.400 + 1.75i)2-s + (−2.02 + 0.974i)4-s + (−1.40 − 1.75i)8-s + (−0.623 + 0.781i)9-s + (−0.678 − 0.541i)11-s + (1.12 − 1.40i)16-s + (−1.62 − 0.781i)18-s + (0.678 − 1.40i)22-s + (−0.846 − 0.193i)23-s − 25-s + (−1.62 + 0.781i)29-s + (0.900 + 0.433i)32-s + (0.5 − 2.19i)36-s + (−0.400 − 1.75i)37-s + (0.0990 + 0.433i)43-s + (1.90 + 0.433i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3479\)    =    \(7^{2} \cdot 71\)
Sign: $0.368 + 0.929i$
Analytic conductor: \(1.73624\)
Root analytic conductor: \(1.31766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3479} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3479,\ (\ :0),\ 0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3127813506\)
\(L(\frac12)\) \(\approx\) \(0.3127813506\)
\(L(1)\) 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 \( 1 + (0.222 + 0.974i)T \)
good2 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
3 \( 1 + (0.623 - 0.781i)T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.846 + 0.193i)T + (0.900 + 0.433i)T^{2} \)
29 \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.222 - 0.974i)T^{2} \)
37 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.900 + 0.433i)T^{2} \)
79 \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.222 + 0.974i)T^{2} \)
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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

−8.933283137412613038259940063148, −8.391173669721937816345478447433, −7.60386008384647696901977554000, −7.38535586312783578234183846641, −6.17554327132081533962671636665, −5.69641940988472642554473979236, −5.19732695879820804635235713204, −4.24816952159144144917680220250, −3.44400305551050215064272721348, −2.19390987259617518107973245289, 0.14988419013853458882722073454, 1.68999589125439346804391166113, 2.42212037504017540819117806438, 3.40677883811771216504954886130, 3.92340205981661260774166535316, 4.89948719733733781113706612591, 5.59048652141687898773819161541, 6.42667731118143177927120851389, 7.65400639721897961145318289035, 8.398151326424044030546571114982

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