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Properties

Label 2-4014-669.668-c1-0-30
Degree $2$
Conductor $4014$
Sign $0.961 - 0.273i$
Analytic cond. $32.0519$
Root an. cond. $5.66144$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + 0.707·5-s + 0.916·7-s + i·8-s − 0.707i·10-s + 4.30·11-s + 4.29i·13-s − 0.916i·14-s + 16-s + 2.79i·17-s + 6.09·19-s − 0.707·20-s − 4.30i·22-s + 0.682·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.316·5-s + 0.346·7-s + 0.353i·8-s − 0.223i·10-s + 1.29·11-s + 1.18i·13-s − 0.244i·14-s + 0.250·16-s + 0.677i·17-s + 1.39·19-s − 0.158·20-s − 0.918i·22-s + 0.142·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4014\)    =    \(2 \cdot 3^{2} \cdot 223\)
Sign: $0.961 - 0.273i$
Analytic conductor: \(32.0519\)
Root analytic conductor: \(5.66144\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4014} (4013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4014,\ (\ :1/2),\ 0.961 - 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.046771710\)
\(L(\frac12)\) \(\approx\) \(2.046771710\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
223 \( 1 + (14.0 + 4.96i)T \)
good5 \( 1 - 0.707T + 5T^{2} \)
7 \( 1 - 0.916T + 7T^{2} \)
11 \( 1 - 4.30T + 11T^{2} \)
13 \( 1 - 4.29iT - 13T^{2} \)
17 \( 1 - 2.79iT - 17T^{2} \)
19 \( 1 - 6.09T + 19T^{2} \)
23 \( 1 - 0.682T + 23T^{2} \)
29 \( 1 - 3.21iT - 29T^{2} \)
31 \( 1 + 0.315T + 31T^{2} \)
37 \( 1 + 4.22T + 37T^{2} \)
41 \( 1 + 0.607iT - 41T^{2} \)
43 \( 1 - 1.08T + 43T^{2} \)
47 \( 1 - 2.90iT - 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 - 0.693T + 59T^{2} \)
61 \( 1 - 9.12iT - 61T^{2} \)
67 \( 1 - 12.9iT - 67T^{2} \)
71 \( 1 + 7.15T + 71T^{2} \)
73 \( 1 + 2.17T + 73T^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 + 0.529iT - 83T^{2} \)
89 \( 1 - 5.43iT - 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−8.923474339983291826186776726120, −7.76764036535357307372538753974, −7.04082082217577322935944272735, −6.20022508390551952433343771204, −5.46420381919877421896112906879, −4.46389607295087990753414112090, −3.90595221913391317439432861010, −2.99484539758198951990562492007, −1.77158841777488061518478181529, −1.28463281702921718726367289223, 0.63887401064968053681407973455, 1.75806889531305819146508182058, 3.12091847243237473377963957413, 3.83093140557343023741251803570, 4.92198927711552096763073683248, 5.43950657960169093562569719426, 6.23012290730220186433044285990, 6.94145539803426972931696777558, 7.70707268000798886578183998605, 8.234353831007742984579860668217

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