Properties

Label 2-4140-23.22-c2-0-48
Degree $2$
Conductor $4140$
Sign $0.229 + 0.973i$
Analytic cond. $112.806$
Root an. cond. $10.6210$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23i·5-s + 13.2i·7-s − 11.2i·11-s − 17.4·13-s + 22.0i·17-s − 6.38i·19-s + (−22.3 + 5.27i)23-s − 5.00·25-s + 26.6·29-s − 17.2·31-s + 29.5·35-s − 33.5i·37-s + 9.39·41-s + 34.5i·43-s + 0.525·47-s + ⋯
L(s)  = 1  − 0.447i·5-s + 1.88i·7-s − 1.01i·11-s − 1.34·13-s + 1.29i·17-s − 0.336i·19-s + (−0.973 + 0.229i)23-s − 0.200·25-s + 0.919·29-s − 0.556·31-s + 0.843·35-s − 0.906i·37-s + 0.229·41-s + 0.802i·43-s + 0.0111·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(112.806\)
Root analytic conductor: \(10.6210\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8694860064\)
\(L(\frac12)\) \(\approx\) \(0.8694860064\)
\(L(2)\) 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 \( 1 + 2.23iT \)
23 \( 1 + (22.3 - 5.27i)T \)
good7 \( 1 - 13.2iT - 49T^{2} \)
11 \( 1 + 11.2iT - 121T^{2} \)
13 \( 1 + 17.4T + 169T^{2} \)
17 \( 1 - 22.0iT - 289T^{2} \)
19 \( 1 + 6.38iT - 361T^{2} \)
29 \( 1 - 26.6T + 841T^{2} \)
31 \( 1 + 17.2T + 961T^{2} \)
37 \( 1 + 33.5iT - 1.36e3T^{2} \)
41 \( 1 - 9.39T + 1.68e3T^{2} \)
43 \( 1 - 34.5iT - 1.84e3T^{2} \)
47 \( 1 - 0.525T + 2.20e3T^{2} \)
53 \( 1 - 1.66iT - 2.80e3T^{2} \)
59 \( 1 - 12.8T + 3.48e3T^{2} \)
61 \( 1 + 74.9iT - 3.72e3T^{2} \)
67 \( 1 - 42.5iT - 4.48e3T^{2} \)
71 \( 1 - 30.9T + 5.04e3T^{2} \)
73 \( 1 + 21.1T + 5.32e3T^{2} \)
79 \( 1 + 25.5iT - 6.24e3T^{2} \)
83 \( 1 + 31.7iT - 6.88e3T^{2} \)
89 \( 1 + 90.7iT - 7.92e3T^{2} \)
97 \( 1 + 51.5iT - 9.40e3T^{2} \)
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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

−8.346195794681178370172532643692, −7.52621168015482059708881903049, −6.34993933425056792137042591895, −5.84313918970145238774376820995, −5.26138300869474778724391767800, −4.42785595246628402917508818725, −3.31770787076689900341944402201, −2.47322203512708318253962284553, −1.75140800210768375975706248863, −0.21830866603103646093841555258, 0.801048355623966963887679441538, 2.02253947683168875219673364086, 2.94320004422105069310327538028, 3.97053313748866558290762888057, 4.54928745236107129515727334185, 5.23500261790077419101559244588, 6.53937060606855480658611071274, 7.07018423411895354008484299268, 7.48470003019078070742985603791, 8.108018299604238887132707566981

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