Properties

Label 2-4140-15.2-c1-0-15
Degree $2$
Conductor $4140$
Sign $-0.274 - 0.961i$
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  + (2.01 + 0.963i)5-s + (2.04 + 2.04i)7-s + 4.66i·11-s + (−0.347 + 0.347i)13-s + (1.10 − 1.10i)17-s + 4.66i·19-s + (0.707 + 0.707i)23-s + (3.14 + 3.88i)25-s − 4.98·29-s + 1.77·31-s + (2.15 + 6.10i)35-s + (−1.84 − 1.84i)37-s + 0.953i·41-s + (−2.26 + 2.26i)43-s + (2.18 − 2.18i)47-s + ⋯
L(s)  = 1  + (0.902 + 0.430i)5-s + (0.773 + 0.773i)7-s + 1.40i·11-s + (−0.0963 + 0.0963i)13-s + (0.267 − 0.267i)17-s + 1.06i·19-s + (0.147 + 0.147i)23-s + (0.628 + 0.777i)25-s − 0.925·29-s + 0.319·31-s + (0.364 + 1.03i)35-s + (−0.303 − 0.303i)37-s + 0.148i·41-s + (−0.346 + 0.346i)43-s + (0.318 − 0.318i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.274 - 0.961i$
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: \((2,\ 4140,\ (\ :1/2),\ -0.274 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.329059375\)
\(L(\frac12)\) \(\approx\) \(2.329059375\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.01 - 0.963i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-2.04 - 2.04i)T + 7iT^{2} \)
11 \( 1 - 4.66iT - 11T^{2} \)
13 \( 1 + (0.347 - 0.347i)T - 13iT^{2} \)
17 \( 1 + (-1.10 + 1.10i)T - 17iT^{2} \)
19 \( 1 - 4.66iT - 19T^{2} \)
29 \( 1 + 4.98T + 29T^{2} \)
31 \( 1 - 1.77T + 31T^{2} \)
37 \( 1 + (1.84 + 1.84i)T + 37iT^{2} \)
41 \( 1 - 0.953iT - 41T^{2} \)
43 \( 1 + (2.26 - 2.26i)T - 43iT^{2} \)
47 \( 1 + (-2.18 + 2.18i)T - 47iT^{2} \)
53 \( 1 + (-2.24 - 2.24i)T + 53iT^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 3.51T + 61T^{2} \)
67 \( 1 + (0.135 + 0.135i)T + 67iT^{2} \)
71 \( 1 + 4.66iT - 71T^{2} \)
73 \( 1 + (-4.35 + 4.35i)T - 73iT^{2} \)
79 \( 1 - 8.72iT - 79T^{2} \)
83 \( 1 + (-2.66 - 2.66i)T + 83iT^{2} \)
89 \( 1 + 3.24T + 89T^{2} \)
97 \( 1 + (8.18 + 8.18i)T + 97iT^{2} \)
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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

−8.685439542659504356426301077435, −7.81679601056683970159750013957, −7.21815985121021651275028483487, −6.39668515148360430292581099342, −5.57932102096412520467757002291, −5.08292878455232252538958644048, −4.17594849970577417491362299902, −3.03084253544786111739160617054, −2.05774303571324522592172614338, −1.60455542296230114613435348473, 0.64440881964912378731488224032, 1.50881954658035785223455433830, 2.61462945040087157527742579344, 3.58016848370682605165641730970, 4.55455392555333599586174267579, 5.23450390248117204037194694772, 5.92244335354659717357638236834, 6.67735380857688203132008529518, 7.54401226481915269062413811835, 8.288792568556367460227318846276

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