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Properties

Label 2-4140-15.2-c1-0-32
Degree $2$
Conductor $4140$
Sign $0.247 + 0.968i$
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  + (1.99 + 1.01i)5-s + (−1.87 − 1.87i)7-s + 1.74i·11-s + (3.99 − 3.99i)13-s + (0.582 − 0.582i)17-s − 7.02i·19-s + (−0.707 − 0.707i)23-s + (2.92 + 4.05i)25-s + 4.66·29-s + 2.39·31-s + (−1.82 − 5.64i)35-s + (−7.42 − 7.42i)37-s + 5.25i·41-s + (−4.98 + 4.98i)43-s + (−7.04 + 7.04i)47-s + ⋯
L(s)  = 1  + (0.889 + 0.456i)5-s + (−0.709 − 0.709i)7-s + 0.524i·11-s + (1.10 − 1.10i)13-s + (0.141 − 0.141i)17-s − 1.61i·19-s + (−0.147 − 0.147i)23-s + (0.584 + 0.811i)25-s + 0.865·29-s + 0.430·31-s + (−0.307 − 0.954i)35-s + (−1.22 − 1.22i)37-s + 0.820i·41-s + (−0.760 + 0.760i)43-s + (−1.02 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.247 + 0.968i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.890616189\)
\(L(\frac12)\) \(\approx\) \(1.890616189\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 - 1.01i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (1.87 + 1.87i)T + 7iT^{2} \)
11 \( 1 - 1.74iT - 11T^{2} \)
13 \( 1 + (-3.99 + 3.99i)T - 13iT^{2} \)
17 \( 1 + (-0.582 + 0.582i)T - 17iT^{2} \)
19 \( 1 + 7.02iT - 19T^{2} \)
29 \( 1 - 4.66T + 29T^{2} \)
31 \( 1 - 2.39T + 31T^{2} \)
37 \( 1 + (7.42 + 7.42i)T + 37iT^{2} \)
41 \( 1 - 5.25iT - 41T^{2} \)
43 \( 1 + (4.98 - 4.98i)T - 43iT^{2} \)
47 \( 1 + (7.04 - 7.04i)T - 47iT^{2} \)
53 \( 1 + (7.72 + 7.72i)T + 53iT^{2} \)
59 \( 1 + 7.38T + 59T^{2} \)
61 \( 1 - 0.462T + 61T^{2} \)
67 \( 1 + (-5.30 - 5.30i)T + 67iT^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-10.1 + 10.1i)T - 73iT^{2} \)
79 \( 1 + 4.66iT - 79T^{2} \)
83 \( 1 + (1.85 + 1.85i)T + 83iT^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (10.6 + 10.6i)T + 97iT^{2} \)
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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

−8.242103151972199407357868357857, −7.44707312679483775002950316912, −6.51717549387878613773443702420, −6.40589165602431602022361048279, −5.28125310054877468975579916877, −4.59258657082222751562493634462, −3.34888374095653655047557625439, −2.95606698981843755022179024013, −1.72645204851156024857871739542, −0.54732745174370246142790624225, 1.27324681067730697341601600491, 2.02789023345739766665747526491, 3.18090652464821475200060528900, 3.88984336241929535455176138910, 4.99628888719642398342949018914, 5.74921776865457850864779909782, 6.31896339530493352062015963544, 6.76705298574267095180206979032, 8.237423634890503200094484146660, 8.508996047445695406638325539133

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