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Properties

Label 2-4140-15.2-c1-0-35
Degree $2$
Conductor $4140$
Sign $-0.146 + 0.989i$
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  + (−1.13 + 1.92i)5-s + (2.43 + 2.43i)7-s − 5.50i·11-s + (1.03 − 1.03i)13-s + (0.543 − 0.543i)17-s − 5.88i·19-s + (0.707 + 0.707i)23-s + (−2.43 − 4.36i)25-s − 7.99·29-s + 3.34·31-s + (−7.43 + 1.93i)35-s + (−3.41 − 3.41i)37-s − 4.57i·41-s + (−1.70 + 1.70i)43-s + (−7.40 + 7.40i)47-s + ⋯
L(s)  = 1  + (−0.506 + 0.862i)5-s + (0.918 + 0.918i)7-s − 1.65i·11-s + (0.286 − 0.286i)13-s + (0.131 − 0.131i)17-s − 1.34i·19-s + (0.147 + 0.147i)23-s + (−0.487 − 0.872i)25-s − 1.48·29-s + 0.601·31-s + (−1.25 + 0.327i)35-s + (−0.561 − 0.561i)37-s − 0.714i·41-s + (−0.259 + 0.259i)43-s + (−1.08 + 1.08i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.146 + 0.989i$
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: \((2,\ 4140,\ (\ :1/2),\ -0.146 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.001524912\)
\(L(\frac12)\) \(\approx\) \(1.001524912\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 - 1.92i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-2.43 - 2.43i)T + 7iT^{2} \)
11 \( 1 + 5.50iT - 11T^{2} \)
13 \( 1 + (-1.03 + 1.03i)T - 13iT^{2} \)
17 \( 1 + (-0.543 + 0.543i)T - 17iT^{2} \)
19 \( 1 + 5.88iT - 19T^{2} \)
29 \( 1 + 7.99T + 29T^{2} \)
31 \( 1 - 3.34T + 31T^{2} \)
37 \( 1 + (3.41 + 3.41i)T + 37iT^{2} \)
41 \( 1 + 4.57iT - 41T^{2} \)
43 \( 1 + (1.70 - 1.70i)T - 43iT^{2} \)
47 \( 1 + (7.40 - 7.40i)T - 47iT^{2} \)
53 \( 1 + (4.73 + 4.73i)T + 53iT^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-3.06 - 3.06i)T + 67iT^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (2.22 - 2.22i)T - 73iT^{2} \)
79 \( 1 + 6.61iT - 79T^{2} \)
83 \( 1 + (3.00 + 3.00i)T + 83iT^{2} \)
89 \( 1 + 2.16T + 89T^{2} \)
97 \( 1 + (-11.3 - 11.3i)T + 97iT^{2} \)
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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

−8.124158963785352657819156539314, −7.68499523209798046227740976426, −6.70879649280050592931716972940, −5.97556046567401295580604451213, −5.35011814970457124674824667941, −4.45945912139311544143459652770, −3.32056986735026743575185374273, −2.90372637993042962241765240188, −1.75560079278084450269792647098, −0.28028961495765392433702718119, 1.38528269745715278852411839276, 1.79750198600275959671228196669, 3.46178554250945748991224038742, 4.27367441860873260185486073738, 4.66903451795766871587849622238, 5.45495053611921856673711653476, 6.53965225740925956105138752981, 7.42838219970367538260895520577, 7.80900494074974520357366169922, 8.435397532014458516434280931191

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