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Properties

Label 2-4140-15.2-c1-0-20
Degree $2$
Conductor $4140$
Sign $0.322 - 0.946i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.322 - 0.946i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.322 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.479635475\)
\(L(\frac12)\) \(\approx\) \(2.479635475\)
\(L(\frac{3}{2})\) 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

−8.386263780193545520645676069736, −8.178275605274300458441766292396, −6.96850497464290092910527939472, −6.20020507407608998823250311816, −5.79290181266439814529864684069, −4.94163857398318890094370872319, −3.94652183043580098972745719553, −2.91825652900350016143307002280, −2.29397629993076422080477954585, −1.21730473498704806510419824673, 0.78136849090636254625380137642, 1.62646720525540389297139816007, 2.52388984893741860917715860395, 4.03219238217636967495795842257, 4.57116243736188848802040466219, 4.97889950566393411797988315806, 6.15634427141817658536500497212, 6.81299094521900080163829890779, 7.58358857931434343813046846704, 8.328088987017115183541538355719

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