Properties

Label 2-4140-5.4-c1-0-32
Degree $2$
Conductor $4140$
Sign $0.732 + 0.680i$
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  + (−1.52 + 1.63i)5-s − 1.80i·7-s + 2.90·11-s − 2.25i·13-s + 2.14i·17-s − 0.339·19-s + i·23-s + (−0.369 − 4.98i)25-s − 5.60·29-s + 5.92·31-s + (2.95 + 2.74i)35-s + 8.98i·37-s − 1.89·41-s − 9.47i·43-s − 7.83i·47-s + ⋯
L(s)  = 1  + (−0.680 + 0.732i)5-s − 0.682i·7-s + 0.876·11-s − 0.624i·13-s + 0.520i·17-s − 0.0779·19-s + 0.208i·23-s + (−0.0738 − 0.997i)25-s − 1.04·29-s + 1.06·31-s + (0.499 + 0.464i)35-s + 1.47i·37-s − 0.295·41-s − 1.44i·43-s − 1.14i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.732 + 0.680i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.732 + 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.473944840\)
\(L(\frac12)\) \(\approx\) \(1.473944840\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.52 - 1.63i)T \)
23 \( 1 - iT \)
good7 \( 1 + 1.80iT - 7T^{2} \)
11 \( 1 - 2.90T + 11T^{2} \)
13 \( 1 + 2.25iT - 13T^{2} \)
17 \( 1 - 2.14iT - 17T^{2} \)
19 \( 1 + 0.339T + 19T^{2} \)
29 \( 1 + 5.60T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 - 8.98iT - 37T^{2} \)
41 \( 1 + 1.89T + 41T^{2} \)
43 \( 1 + 9.47iT - 43T^{2} \)
47 \( 1 + 7.83iT - 47T^{2} \)
53 \( 1 + 6.47iT - 53T^{2} \)
59 \( 1 - 5.17T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 - 9.25iT - 67T^{2} \)
71 \( 1 - 4.60T + 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 7.94T + 79T^{2} \)
83 \( 1 + 5.37iT - 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 2.43iT - 97T^{2} \)
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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

−8.247909338049063202914973104408, −7.55452555275239321089400885276, −6.89009097379058855244734317520, −6.33859229013356331204506925670, −5.37592537049983592706060068387, −4.32463876784982910279422770783, −3.72751562856158620341888133517, −3.05031680884573032933871447537, −1.80628131994884161739779975848, −0.53310790546795136910185305106, 0.928430532343039732031306794517, 2.01051046746176026339970252574, 3.10902490547772127034769617535, 4.10502513252580208626520118982, 4.60333350672764029082833808229, 5.54373895676632044676462886900, 6.26060948952315614431366070530, 7.12032179424065685442520790761, 7.83027610713300746089627486430, 8.566532564546566779201218168567

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