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Properties

Label 2-4140-345.344-c1-0-18
Degree $2$
Conductor $4140$
Sign $0.577 - 0.816i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
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  + 2.23i·5-s − 3.40·7-s − 8.20i·17-s − 4.79i·23-s − 5.00·25-s + 10.0i·29-s + 7.95·31-s − 7.60i·35-s + 1.73·37-s + 3.98i·41-s + 13.1·43-s + 4.56·49-s − 11.3i·53-s + 13.2i·59-s + 8.68·67-s + ⋯
L(s)  = 1  + 0.999i·5-s − 1.28·7-s − 1.99i·17-s − 0.999i·23-s − 1.00·25-s + 1.87i·29-s + 1.42·31-s − 1.28i·35-s + 0.285·37-s + 0.622i·41-s + 1.99·43-s + 0.652·49-s − 1.55i·53-s + 1.71i·59-s + 1.06·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.339829757\)
\(L(\frac12)\) \(\approx\) \(1.339829757\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 2.23iT \)
23 \( 1 + 4.79iT \)
good7 \( 1 + 3.40T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 8.20iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 - 7.95T + 31T^{2} \)
37 \( 1 - 1.73T + 37T^{2} \)
41 \( 1 - 3.98iT - 41T^{2} \)
43 \( 1 - 13.1T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 - 13.2iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 8.68T + 67T^{2} \)
71 \( 1 - 6.99iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 6.86iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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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

−8.603959949757752431727831027104, −7.58540667026737219572635613829, −6.88477822655428578303571668857, −6.59357942368613436347270811265, −5.69509012824152453337940472274, −4.78114170639828505102748324576, −3.78946332702303618907502675141, −2.83552163017422209590146759292, −2.61929613754241521817641031516, −0.78326786043994733520547785553, 0.54019843943366028455361021002, 1.72294990645388852377210066209, 2.82221471335863771958746265100, 3.93332651098257452063488596730, 4.26542833816540306246457515426, 5.55839029100104378630781708832, 6.02882720253262315774383430730, 6.66054023279208378709262040324, 7.87644226502184143668619155735, 8.151298560658309643957067418332

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