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Properties

Label 2-2940-105.104-c1-0-27
Degree $2$
Conductor $2940$
Sign $0.00127 - 0.999i$
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  + (1.73 − 0.0514i)3-s + (−0.977 + 2.01i)5-s + (2.99 − 0.178i)9-s + 3.03i·11-s − 1.07·13-s + (−1.58 + 3.53i)15-s + 1.57i·17-s − 2.22i·19-s + 6.24·23-s + (−3.08 − 3.93i)25-s + (5.17 − 0.462i)27-s + 5.90i·29-s − 5.73i·31-s + (0.156 + 5.26i)33-s + 5.60i·37-s + ⋯
L(s)  = 1  + (0.999 − 0.0296i)3-s + (−0.437 + 0.899i)5-s + (0.998 − 0.0593i)9-s + 0.916i·11-s − 0.298·13-s + (−0.410 + 0.911i)15-s + 0.382i·17-s − 0.511i·19-s + 1.30·23-s + (−0.617 − 0.786i)25-s + (0.996 − 0.0889i)27-s + 1.09i·29-s − 1.02i·31-s + (0.0272 + 0.916i)33-s + 0.920i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00127 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00127 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.00127 - 0.999i$
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: \((2,\ 2940,\ (\ :1/2),\ 0.00127 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.284858642\)
\(L(\frac12)\) \(\approx\) \(2.284858642\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.0514i)T \)
5 \( 1 + (0.977 - 2.01i)T \)
7 \( 1 \)
good11 \( 1 - 3.03iT - 11T^{2} \)
13 \( 1 + 1.07T + 13T^{2} \)
17 \( 1 - 1.57iT - 17T^{2} \)
19 \( 1 + 2.22iT - 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 - 5.90iT - 29T^{2} \)
31 \( 1 + 5.73iT - 31T^{2} \)
37 \( 1 - 5.60iT - 37T^{2} \)
41 \( 1 - 7.52T + 41T^{2} \)
43 \( 1 - 5.71iT - 43T^{2} \)
47 \( 1 - 6.90iT - 47T^{2} \)
53 \( 1 + 9.43T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 - 5.66iT - 67T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + 1.42T + 73T^{2} \)
79 \( 1 + 7.69T + 79T^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 + 1.54T + 89T^{2} \)
97 \( 1 - 1.33T + 97T^{2} \)
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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

−8.977087806159946233561400146381, −8.072621140870130014893450432841, −7.39468285492731209735690047578, −6.99403921525327476932749446815, −6.11002407274426455875308074887, −4.74746376092981094876581050716, −4.20294338144983904500809784843, −3.08303091925248846155891187121, −2.63600081972580956102705919644, −1.42920092110431625999712416986, 0.64867168895656136752093027049, 1.81347253299562540731455337115, 2.99661820936704870149737415152, 3.69156829329291357042673847293, 4.58888321534831897074134916630, 5.29772493212419360867719926551, 6.34287654254118651594727912643, 7.38305793024930602527413137591, 7.87402094917911649206201598379, 8.650456352178281174214132832757

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