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Properties

Label 2-2940-105.104-c1-0-22
Degree $2$
Conductor $2940$
Sign $0.434 - 0.900i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
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  + (1.14 − 1.29i)3-s + (0.933 + 2.03i)5-s + (−0.372 − 2.97i)9-s + 2.01i·11-s − 3.28·13-s + (3.70 + 1.11i)15-s + 1.71i·17-s + 4.80i·19-s + 1.36·23-s + (−3.25 + 3.79i)25-s + (−4.29 − 2.92i)27-s + 2.21i·29-s + 3.56i·31-s + (2.62 + 2.31i)33-s + 11.2i·37-s + ⋯
L(s)  = 1  + (0.661 − 0.749i)3-s + (0.417 + 0.908i)5-s + (−0.124 − 0.992i)9-s + 0.608i·11-s − 0.910·13-s + (0.957 + 0.288i)15-s + 0.416i·17-s + 1.10i·19-s + 0.284·23-s + (−0.651 + 0.758i)25-s + (−0.826 − 0.563i)27-s + 0.412i·29-s + 0.640i·31-s + (0.456 + 0.402i)33-s + 1.84i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.008281141\)
\(L(\frac12)\) \(\approx\) \(2.008281141\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 + 1.29i)T \)
5 \( 1 + (-0.933 - 2.03i)T \)
7 \( 1 \)
good11 \( 1 - 2.01iT - 11T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 - 1.71iT - 17T^{2} \)
19 \( 1 - 4.80iT - 19T^{2} \)
23 \( 1 - 1.36T + 23T^{2} \)
29 \( 1 - 2.21iT - 29T^{2} \)
31 \( 1 - 3.56iT - 31T^{2} \)
37 \( 1 - 11.2iT - 37T^{2} \)
41 \( 1 - 8.97T + 41T^{2} \)
43 \( 1 + 4.39iT - 43T^{2} \)
47 \( 1 + 2.50iT - 47T^{2} \)
53 \( 1 + 2.08T + 53T^{2} \)
59 \( 1 + 1.46T + 59T^{2} \)
61 \( 1 - 12.5iT - 61T^{2} \)
67 \( 1 - 2.94iT - 67T^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 - 9.62T + 79T^{2} \)
83 \( 1 - 14.2iT - 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 0.657T + 97T^{2} \)
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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

−8.862789969717942078849389591531, −7.934248486086143126901159885718, −7.41186904000260455094520336274, −6.71241609067051343878144835540, −6.10657425537355816667003399897, −5.10686353426405500159809025633, −3.93237524666513122681599019460, −3.05748891782098562650392614735, −2.29230875639304842322205736023, −1.42430336304151915579977810173, 0.55894648353203519553872966139, 2.13011159530608620879177679758, 2.83496356539328443617306003354, 3.98680191954094410952427041389, 4.72096822236787388676983925397, 5.31783457299019737068777245676, 6.16602504105289461730117445098, 7.37960600368498545546376197720, 7.962023628845535923267287311175, 8.832021198390066501779331125744

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