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Properties

Label 2-2940-35.13-c1-0-6
Degree $2$
Conductor $2940$
Sign $-0.752 - 0.658i$
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  + (0.707 − 0.707i)3-s + (0.998 + 2.00i)5-s − 1.00i·9-s + 1.30·11-s + (−4.38 + 4.38i)13-s + (2.12 + 0.708i)15-s + (−3.53 − 3.53i)17-s − 2.42·19-s + (−2.53 − 2.53i)23-s + (−3.00 + 3.99i)25-s + (−0.707 − 0.707i)27-s + 7.14i·29-s + 4.34i·31-s + (0.921 − 0.921i)33-s + (−3.23 + 3.23i)37-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.446 + 0.894i)5-s − 0.333i·9-s + 0.392·11-s + (−1.21 + 1.21i)13-s + (0.547 + 0.182i)15-s + (−0.857 − 0.857i)17-s − 0.556·19-s + (−0.529 − 0.529i)23-s + (−0.601 + 0.799i)25-s + (−0.136 − 0.136i)27-s + 1.32i·29-s + 0.781i·31-s + (0.160 − 0.160i)33-s + (−0.531 + 0.531i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.752 - 0.658i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9314472486\)
\(L(\frac12)\) \(\approx\) \(0.9314472486\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.998 - 2.00i)T \)
7 \( 1 \)
good11 \( 1 - 1.30T + 11T^{2} \)
13 \( 1 + (4.38 - 4.38i)T - 13iT^{2} \)
17 \( 1 + (3.53 + 3.53i)T + 17iT^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
23 \( 1 + (2.53 + 2.53i)T + 23iT^{2} \)
29 \( 1 - 7.14iT - 29T^{2} \)
31 \( 1 - 4.34iT - 31T^{2} \)
37 \( 1 + (3.23 - 3.23i)T - 37iT^{2} \)
41 \( 1 - 4.53iT - 41T^{2} \)
43 \( 1 + (6.39 + 6.39i)T + 43iT^{2} \)
47 \( 1 + (1.82 + 1.82i)T + 47iT^{2} \)
53 \( 1 + (-5.52 - 5.52i)T + 53iT^{2} \)
59 \( 1 + 3.07T + 59T^{2} \)
61 \( 1 - 4.88iT - 61T^{2} \)
67 \( 1 + (-5.11 + 5.11i)T - 67iT^{2} \)
71 \( 1 + 1.92T + 71T^{2} \)
73 \( 1 + (-6.67 + 6.67i)T - 73iT^{2} \)
79 \( 1 + 8.45iT - 79T^{2} \)
83 \( 1 + (7.74 - 7.74i)T - 83iT^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + (-4.04 - 4.04i)T + 97iT^{2} \)
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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

−9.058241973940847437799804283087, −8.379657264611985130532286465726, −7.15550449920371566487715556263, −6.95610215127318773891732859742, −6.33100446840275423425497336553, −5.14309644183411294330076665410, −4.34046381398513247071658015753, −3.25323613763329119000891751031, −2.39926334407888088772842588333, −1.70770967548743581853357368126, 0.24591422491474582727851488879, 1.81107285720440029176969393144, 2.58577171249457484652614472326, 3.85820185339297906001422290595, 4.47369786262767376979842667970, 5.36650269591796800481717571105, 5.98098509317220057146134626252, 6.99477167284369983960448978286, 8.107741625888724200756986581342, 8.295229787021210835401279641935

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