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Properties

Label 2-4140-15.2-c1-0-0
Degree $2$
Conductor $4140$
Sign $-0.990 - 0.135i$
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  + (1.94 + 1.11i)5-s + (−0.576 − 0.576i)7-s + 0.279i·11-s + (−1.37 + 1.37i)13-s + (−1.92 + 1.92i)17-s + 4.53i·19-s + (−0.707 − 0.707i)23-s + (2.53 + 4.31i)25-s − 9.30·29-s − 8.69·31-s + (−0.478 − 1.76i)35-s + (−2.34 − 2.34i)37-s − 5.29i·41-s + (5.71 − 5.71i)43-s + (−6.03 + 6.03i)47-s + ⋯
L(s)  = 1  + (0.867 + 0.496i)5-s + (−0.218 − 0.218i)7-s + 0.0843i·11-s + (−0.380 + 0.380i)13-s + (−0.465 + 0.465i)17-s + 1.04i·19-s + (−0.147 − 0.147i)23-s + (0.506 + 0.862i)25-s − 1.72·29-s − 1.56·31-s + (−0.0809 − 0.297i)35-s + (−0.384 − 0.384i)37-s − 0.826i·41-s + (0.871 − 0.871i)43-s + (−0.879 + 0.879i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.990 - 0.135i$
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: \((2,\ 4140,\ (\ :1/2),\ -0.990 - 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5297956384\)
\(L(\frac12)\) \(\approx\) \(0.5297956384\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.11i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (0.576 + 0.576i)T + 7iT^{2} \)
11 \( 1 - 0.279iT - 11T^{2} \)
13 \( 1 + (1.37 - 1.37i)T - 13iT^{2} \)
17 \( 1 + (1.92 - 1.92i)T - 17iT^{2} \)
19 \( 1 - 4.53iT - 19T^{2} \)
29 \( 1 + 9.30T + 29T^{2} \)
31 \( 1 + 8.69T + 31T^{2} \)
37 \( 1 + (2.34 + 2.34i)T + 37iT^{2} \)
41 \( 1 + 5.29iT - 41T^{2} \)
43 \( 1 + (-5.71 + 5.71i)T - 43iT^{2} \)
47 \( 1 + (6.03 - 6.03i)T - 47iT^{2} \)
53 \( 1 + (7.60 + 7.60i)T + 53iT^{2} \)
59 \( 1 - 4.99T + 59T^{2} \)
61 \( 1 + 6.61T + 61T^{2} \)
67 \( 1 + (-7.67 - 7.67i)T + 67iT^{2} \)
71 \( 1 - 8.41iT - 71T^{2} \)
73 \( 1 + (3.26 - 3.26i)T - 73iT^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 + (-2.76 - 2.76i)T + 83iT^{2} \)
89 \( 1 - 5.27T + 89T^{2} \)
97 \( 1 + (8.87 + 8.87i)T + 97iT^{2} \)
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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

−8.966259135753573188022146201084, −7.942168167151651641596901737499, −7.18715460819660752368887615196, −6.63847526620402781539341542002, −5.74781713948759859795525099523, −5.30558406400573342309824350226, −4.04130937042502114866333724218, −3.47653834658132866973731515709, −2.22361819335874980904203844143, −1.69162037128608008376476195135, 0.13344208009578896517534724791, 1.52752180807015214411553776752, 2.43008447041372928358709912415, 3.28056580107650354207404736307, 4.44510396426401498587836353157, 5.15553015187203693291099465513, 5.77257569257068495786032704016, 6.54310948127868766210694105138, 7.32917702316389391364204598810, 8.076642777948927347308987532985

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