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Properties

Label 2-4020-67.29-c1-0-0
Degree $2$
Conductor $4020$
Sign $-0.667 - 0.745i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
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  − 3-s − 5-s + (−0.533 − 0.923i)7-s + 9-s + (−2.16 − 3.75i)11-s + (0.803 − 1.39i)13-s + 15-s + (0.990 − 1.71i)17-s + (−1.89 + 3.27i)19-s + (0.533 + 0.923i)21-s + (−0.666 + 1.15i)23-s + 25-s − 27-s + (0.0815 + 0.141i)29-s + (−2.43 − 4.21i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + (−0.201 − 0.349i)7-s + 0.333·9-s + (−0.653 − 1.13i)11-s + (0.222 − 0.385i)13-s + 0.258·15-s + (0.240 − 0.416i)17-s + (−0.434 + 0.752i)19-s + (0.116 + 0.201i)21-s + (−0.138 + 0.240i)23-s + 0.200·25-s − 0.192·27-s + (0.0151 + 0.0262i)29-s + (−0.437 − 0.757i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.667 - 0.745i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (3781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.667 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05352913965\)
\(L(\frac12)\) \(\approx\) \(0.05352913965\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
67 \( 1 + (-2.49 - 7.79i)T \)
good7 \( 1 + (0.533 + 0.923i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.16 + 3.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.803 + 1.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.990 + 1.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.89 - 3.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.666 - 1.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0815 - 0.141i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.43 + 4.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.247 - 0.429i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.78 + 3.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.04T + 43T^{2} \)
47 \( 1 + (5.89 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 - 7.23T + 59T^{2} \)
61 \( 1 + (4.37 - 7.56i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-0.647 - 1.12i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.34 + 7.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.53 - 13.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.30 - 9.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.177T + 89T^{2} \)
97 \( 1 + (8.71 - 15.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.432475319054543237819814393601, −8.114067073883696046839203988086, −7.17662329789479333114969280288, −6.55394293973481748833148058750, −5.54346814648658337487304165684, −5.28510283602507239669678229579, −3.95841379387612313504477887314, −3.53188101348993278395148499304, −2.37563210252556507046206735657, −0.959671060046606467110433058791, 0.02006117278876020080235248015, 1.55463780962614123316795919075, 2.55557594409608360672663181344, 3.60666228709984561798005625619, 4.58241099178801995729021408411, 5.02740159611002346513123639131, 6.02215098808648671897411367131, 6.72842775266882996567234215230, 7.34798936623582322768713736691, 8.140350326830611757752276858673

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