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Properties

Label 2-4140-15.2-c1-0-25
Degree $2$
Conductor $4140$
Sign $0.979 - 0.199i$
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  + (2.19 + 0.406i)5-s + (1.26 + 1.26i)7-s − 3.93i·11-s + (1.46 − 1.46i)13-s + (−0.467 + 0.467i)17-s + 4.88i·19-s + (0.707 + 0.707i)23-s + (4.66 + 1.78i)25-s − 0.536·29-s + 3.91·31-s + (2.26 + 3.29i)35-s + (−0.291 − 0.291i)37-s + 11.7i·41-s + (−2.67 + 2.67i)43-s + (7.55 − 7.55i)47-s + ⋯
L(s)  = 1  + (0.983 + 0.181i)5-s + (0.477 + 0.477i)7-s − 1.18i·11-s + (0.407 − 0.407i)13-s + (−0.113 + 0.113i)17-s + 1.12i·19-s + (0.147 + 0.147i)23-s + (0.933 + 0.357i)25-s − 0.0996·29-s + 0.703·31-s + (0.382 + 0.556i)35-s + (−0.0480 − 0.0480i)37-s + 1.83i·41-s + (−0.407 + 0.407i)43-s + (1.10 − 1.10i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.979 - 0.199i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.979 - 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.641327222\)
\(L(\frac12)\) \(\approx\) \(2.641327222\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 - 0.406i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-1.26 - 1.26i)T + 7iT^{2} \)
11 \( 1 + 3.93iT - 11T^{2} \)
13 \( 1 + (-1.46 + 1.46i)T - 13iT^{2} \)
17 \( 1 + (0.467 - 0.467i)T - 17iT^{2} \)
19 \( 1 - 4.88iT - 19T^{2} \)
29 \( 1 + 0.536T + 29T^{2} \)
31 \( 1 - 3.91T + 31T^{2} \)
37 \( 1 + (0.291 + 0.291i)T + 37iT^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 + (2.67 - 2.67i)T - 43iT^{2} \)
47 \( 1 + (-7.55 + 7.55i)T - 47iT^{2} \)
53 \( 1 + (0.0492 + 0.0492i)T + 53iT^{2} \)
59 \( 1 - 9.00T + 59T^{2} \)
61 \( 1 - 3.92T + 61T^{2} \)
67 \( 1 + (-0.268 - 0.268i)T + 67iT^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (4.49 - 4.49i)T - 73iT^{2} \)
79 \( 1 + 1.65iT - 79T^{2} \)
83 \( 1 + (-5.31 - 5.31i)T + 83iT^{2} \)
89 \( 1 + 6.48T + 89T^{2} \)
97 \( 1 + (-11.1 - 11.1i)T + 97iT^{2} \)
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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

−8.416970851967830252477726161085, −7.952678005324901108434318028388, −6.80386490538908542356425995359, −6.07047735111722510210017684857, −5.63742881039191275205251959693, −4.89431134669973445936055440452, −3.70481345352187946559868549161, −2.93391961862481997530588241981, −1.98046781915486808758994331298, −1.01224403945108612084281076184, 0.936666562023955568466794683756, 1.95294968540928610646941997993, 2.65512211111110488147177005426, 4.01232180228342815585300014997, 4.67548709195671752549294511633, 5.32773454297188091884998803950, 6.24013170281611534919836074851, 7.01340400408786504283218709395, 7.44999334039593305102887599240, 8.623459902047652614706944063955

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