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Properties

Label 2-4140-15.8-c1-0-42
Degree $2$
Conductor $4140$
Sign $-0.924 - 0.380i$
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  + (−2.23 − 0.0138i)5-s + (2.36 − 2.36i)7-s − 6.45i·11-s + (−1.76 − 1.76i)13-s + (−4.25 − 4.25i)17-s + 2.13i·19-s + (0.707 − 0.707i)23-s + (4.99 + 0.0619i)25-s − 4.27·29-s − 7.22·31-s + (−5.31 + 5.25i)35-s + (−2.32 + 2.32i)37-s − 4.23i·41-s + (3.77 + 3.77i)43-s + (−3.19 − 3.19i)47-s + ⋯
L(s)  = 1  + (−0.999 − 0.00619i)5-s + (0.893 − 0.893i)7-s − 1.94i·11-s + (−0.489 − 0.489i)13-s + (−1.03 − 1.03i)17-s + 0.490i·19-s + (0.147 − 0.147i)23-s + (0.999 + 0.0123i)25-s − 0.794·29-s − 1.29·31-s + (−0.899 + 0.888i)35-s + (−0.381 + 0.381i)37-s − 0.660i·41-s + (0.576 + 0.576i)43-s + (−0.465 − 0.465i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.924 - 0.380i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.924 - 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5834878866\)
\(L(\frac12)\) \(\approx\) \(0.5834878866\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0138i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-2.36 + 2.36i)T - 7iT^{2} \)
11 \( 1 + 6.45iT - 11T^{2} \)
13 \( 1 + (1.76 + 1.76i)T + 13iT^{2} \)
17 \( 1 + (4.25 + 4.25i)T + 17iT^{2} \)
19 \( 1 - 2.13iT - 19T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 + 7.22T + 31T^{2} \)
37 \( 1 + (2.32 - 2.32i)T - 37iT^{2} \)
41 \( 1 + 4.23iT - 41T^{2} \)
43 \( 1 + (-3.77 - 3.77i)T + 43iT^{2} \)
47 \( 1 + (3.19 + 3.19i)T + 47iT^{2} \)
53 \( 1 + (-5.04 + 5.04i)T - 53iT^{2} \)
59 \( 1 - 7.56T + 59T^{2} \)
61 \( 1 + 1.07T + 61T^{2} \)
67 \( 1 + (5.84 - 5.84i)T - 67iT^{2} \)
71 \( 1 - 6.64iT - 71T^{2} \)
73 \( 1 + (-7.79 - 7.79i)T + 73iT^{2} \)
79 \( 1 - 10.0iT - 79T^{2} \)
83 \( 1 + (10.5 - 10.5i)T - 83iT^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (-8.04 + 8.04i)T - 97iT^{2} \)
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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

−8.047894375394371583515447217971, −7.33704999265645335152217037235, −6.79231804525893023669284038149, −5.60214992879196555599510709018, −5.03081749974225508835467161148, −4.03984761876854682763983920637, −3.54368521867414241861449237230, −2.52363274763361278902909905091, −1.04651202199875495275433458459, −0.18407390688251071309293025475, 1.81439699092475413660924189358, 2.21868812294576749840643487593, 3.59227885438351478084604732141, 4.58643521445268626678541146279, 4.73308964673788350199396612873, 5.82015618214798329973282325680, 6.90786872490008715012603170396, 7.37372120688884325135307505777, 7.995955648896665196019110225145, 8.987609674370468246800888893940

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