Properties

Label 2-4140-15.2-c1-0-33
Degree $2$
Conductor $4140$
Sign $-0.507 + 0.861i$
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  + (−0.692 − 2.12i)5-s + (−0.357 − 0.357i)7-s − 5.38i·11-s + (3.74 − 3.74i)13-s + (3.16 − 3.16i)17-s − 1.26i·19-s + (0.707 + 0.707i)23-s + (−4.04 + 2.94i)25-s + 7.94·29-s + 10.7·31-s + (−0.512 + 1.00i)35-s + (7.93 + 7.93i)37-s − 0.157i·41-s + (−6.80 + 6.80i)43-s + (3.07 − 3.07i)47-s + ⋯
L(s)  = 1  + (−0.309 − 0.950i)5-s + (−0.135 − 0.135i)7-s − 1.62i·11-s + (1.04 − 1.04i)13-s + (0.768 − 0.768i)17-s − 0.290i·19-s + (0.147 + 0.147i)23-s + (−0.808 + 0.588i)25-s + 1.47·29-s + 1.93·31-s + (−0.0867 + 0.170i)35-s + (1.30 + 1.30i)37-s − 0.0245i·41-s + (−1.03 + 1.03i)43-s + (0.448 − 0.448i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.507 + 0.861i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.507 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.901799418\)
\(L(\frac12)\) \(\approx\) \(1.901799418\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.692 + 2.12i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (0.357 + 0.357i)T + 7iT^{2} \)
11 \( 1 + 5.38iT - 11T^{2} \)
13 \( 1 + (-3.74 + 3.74i)T - 13iT^{2} \)
17 \( 1 + (-3.16 + 3.16i)T - 17iT^{2} \)
19 \( 1 + 1.26iT - 19T^{2} \)
29 \( 1 - 7.94T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + (-7.93 - 7.93i)T + 37iT^{2} \)
41 \( 1 + 0.157iT - 41T^{2} \)
43 \( 1 + (6.80 - 6.80i)T - 43iT^{2} \)
47 \( 1 + (-3.07 + 3.07i)T - 47iT^{2} \)
53 \( 1 + (5.49 + 5.49i)T + 53iT^{2} \)
59 \( 1 + 6.30T + 59T^{2} \)
61 \( 1 + 9.56T + 61T^{2} \)
67 \( 1 + (7.26 + 7.26i)T + 67iT^{2} \)
71 \( 1 + 0.286iT - 71T^{2} \)
73 \( 1 + (-0.811 + 0.811i)T - 73iT^{2} \)
79 \( 1 - 9.01iT - 79T^{2} \)
83 \( 1 + (3.24 + 3.24i)T + 83iT^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + (-5.54 - 5.54i)T + 97iT^{2} \)
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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

−8.210841484877613094989783591297, −7.81498159142303640587063054626, −6.45569789227411949324993712356, −6.03936839171274693669799158916, −5.11772109405440288212720565978, −4.52733467699001766631045522445, −3.28102069107421193039278012258, −3.03131036391427813190813449157, −1.13265642951609928928058662208, −0.67304024631030393689374525587, 1.34837165300377799624008534323, 2.34591034154854839359495040269, 3.22190219024981997162739892302, 4.21399201393285280724814830843, 4.62653651173156338510146868785, 6.18484331907388486443667072096, 6.25373093208977224970639659887, 7.26302128359577810090699559739, 7.78175693509615353229482249218, 8.619118796929430908608194762850

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