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Properties

Label 2-4140-15.2-c1-0-43
Degree $2$
Conductor $4140$
Sign $-0.370 - 0.928i$
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  + (0.00942 − 2.23i)5-s + (−1.55 − 1.55i)7-s − 6.24i·11-s + (−4.66 + 4.66i)13-s + (2.07 − 2.07i)17-s + 3.70i·19-s + (0.707 + 0.707i)23-s + (−4.99 − 0.0421i)25-s − 7.12·29-s − 2.07·31-s + (−3.48 + 3.45i)35-s + (−1.56 − 1.56i)37-s + 0.905i·41-s + (5.14 − 5.14i)43-s + (−1.35 + 1.35i)47-s + ⋯
L(s)  = 1  + (0.00421 − 0.999i)5-s + (−0.587 − 0.587i)7-s − 1.88i·11-s + (−1.29 + 1.29i)13-s + (0.503 − 0.503i)17-s + 0.849i·19-s + (0.147 + 0.147i)23-s + (−0.999 − 0.00843i)25-s − 1.32·29-s − 0.373·31-s + (−0.589 + 0.584i)35-s + (−0.257 − 0.257i)37-s + 0.141i·41-s + (0.784 − 0.784i)43-s + (−0.198 + 0.198i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.370 - 0.928i$
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: \((2,\ 4140,\ (\ :1/2),\ -0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.007683434842\)
\(L(\frac12)\) \(\approx\) \(0.007683434842\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.00942 + 2.23i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (1.55 + 1.55i)T + 7iT^{2} \)
11 \( 1 + 6.24iT - 11T^{2} \)
13 \( 1 + (4.66 - 4.66i)T - 13iT^{2} \)
17 \( 1 + (-2.07 + 2.07i)T - 17iT^{2} \)
19 \( 1 - 3.70iT - 19T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
31 \( 1 + 2.07T + 31T^{2} \)
37 \( 1 + (1.56 + 1.56i)T + 37iT^{2} \)
41 \( 1 - 0.905iT - 41T^{2} \)
43 \( 1 + (-5.14 + 5.14i)T - 43iT^{2} \)
47 \( 1 + (1.35 - 1.35i)T - 47iT^{2} \)
53 \( 1 + (-7.81 - 7.81i)T + 53iT^{2} \)
59 \( 1 + 9.74T + 59T^{2} \)
61 \( 1 + 1.55T + 61T^{2} \)
67 \( 1 + (3.97 + 3.97i)T + 67iT^{2} \)
71 \( 1 - 2.59iT - 71T^{2} \)
73 \( 1 + (-5.20 + 5.20i)T - 73iT^{2} \)
79 \( 1 - 1.96iT - 79T^{2} \)
83 \( 1 + (-8.81 - 8.81i)T + 83iT^{2} \)
89 \( 1 - 1.58T + 89T^{2} \)
97 \( 1 + (-5.50 - 5.50i)T + 97iT^{2} \)
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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

−7.80057906863073349774311962476, −7.36607228578702702593345802736, −6.34619131605977569480826562407, −5.66056924736273164817686006062, −4.97964253693811439418671610881, −3.97088158691620683350739252522, −3.45010129737039040483926591884, −2.20827440930642702841743355210, −1.03067157237220345710027644336, −0.00234907191960093759431698057, 1.94998796883829254366856061263, 2.63871882004931202554260093291, 3.35421015727117174890115278744, 4.45034500791460523411775686717, 5.26847340658525640031720513562, 5.98203833633470622734917033382, 6.91644565346697311897088502841, 7.38657068071086803969005995333, 7.904050164572948180731547962813, 9.146373928043912973032512539508

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