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Properties

Label 2-4140-15.2-c1-0-27
Degree $2$
Conductor $4140$
Sign $0.784 + 0.619i$
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  + (−1.10 − 1.94i)5-s + (2.66 + 2.66i)7-s + 2.33i·11-s + (3.05 − 3.05i)13-s + (4.99 − 4.99i)17-s − 1.35i·19-s + (−0.707 − 0.707i)23-s + (−2.53 + 4.30i)25-s − 9.20·29-s + 0.798·31-s + (2.21 − 8.12i)35-s + (4.61 + 4.61i)37-s − 0.542i·41-s + (7.47 − 7.47i)43-s + (5.82 − 5.82i)47-s + ⋯
L(s)  = 1  + (−0.496 − 0.868i)5-s + (1.00 + 1.00i)7-s + 0.705i·11-s + (0.846 − 0.846i)13-s + (1.21 − 1.21i)17-s − 0.310i·19-s + (−0.147 − 0.147i)23-s + (−0.507 + 0.861i)25-s − 1.70·29-s + 0.143·31-s + (0.374 − 1.37i)35-s + (0.758 + 0.758i)37-s − 0.0847i·41-s + (1.14 − 1.14i)43-s + (0.849 − 0.849i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.784 + 0.619i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.784 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.078260942\)
\(L(\frac12)\) \(\approx\) \(2.078260942\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 + 1.94i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-2.66 - 2.66i)T + 7iT^{2} \)
11 \( 1 - 2.33iT - 11T^{2} \)
13 \( 1 + (-3.05 + 3.05i)T - 13iT^{2} \)
17 \( 1 + (-4.99 + 4.99i)T - 17iT^{2} \)
19 \( 1 + 1.35iT - 19T^{2} \)
29 \( 1 + 9.20T + 29T^{2} \)
31 \( 1 - 0.798T + 31T^{2} \)
37 \( 1 + (-4.61 - 4.61i)T + 37iT^{2} \)
41 \( 1 + 0.542iT - 41T^{2} \)
43 \( 1 + (-7.47 + 7.47i)T - 43iT^{2} \)
47 \( 1 + (-5.82 + 5.82i)T - 47iT^{2} \)
53 \( 1 + (-3.20 - 3.20i)T + 53iT^{2} \)
59 \( 1 + 9.30T + 59T^{2} \)
61 \( 1 - 5.23T + 61T^{2} \)
67 \( 1 + (-1.04 - 1.04i)T + 67iT^{2} \)
71 \( 1 + 6.54iT - 71T^{2} \)
73 \( 1 + (9.71 - 9.71i)T - 73iT^{2} \)
79 \( 1 - 2.16iT - 79T^{2} \)
83 \( 1 + (-4.11 - 4.11i)T + 83iT^{2} \)
89 \( 1 + 5.23T + 89T^{2} \)
97 \( 1 + (-5.05 - 5.05i)T + 97iT^{2} \)
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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

−8.283086795493553349169144128128, −7.74805613597312985621187292644, −7.14755813744087441461914544908, −5.63946302021662885577242539113, −5.52919811332073959923662556985, −4.67621855987253572774324653364, −3.84716066531394460315972977572, −2.79199025847410641256914891480, −1.76031493907921117935964112328, −0.74134184952799588504155563639, 1.01831833634966043268012309812, 1.93529382837864613507010610841, 3.30880294022455031734650884734, 3.89565173220658970635352609343, 4.44370280457103578632757080056, 5.83710844931970452610410335885, 6.12786816557323141469655106902, 7.32623765196982499156364668798, 7.66538623886531346763651127388, 8.271269309827669445461180764823

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