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Properties

Label 2-4020-67.37-c1-0-16
Degree $2$
Conductor $4020$
Sign $0.553 - 0.832i$
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 + (−0.388 + 0.673i)7-s + 9-s + (−2.02 + 3.50i)11-s + (−1.61 − 2.79i)13-s + 15-s + (1.72 + 2.98i)17-s + (0.784 + 1.35i)19-s + (−0.388 + 0.673i)21-s + (−0.0608 − 0.105i)23-s + 25-s + 27-s + (4.58 − 7.94i)29-s + (3.20 − 5.55i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (−0.146 + 0.254i)7-s + 0.333·9-s + (−0.609 + 1.05i)11-s + (−0.447 − 0.775i)13-s + 0.258·15-s + (0.418 + 0.724i)17-s + (0.179 + 0.311i)19-s + (−0.0848 + 0.146i)21-s + (−0.0126 − 0.0219i)23-s + 0.200·25-s + 0.192·27-s + (0.851 − 1.47i)29-s + (0.575 − 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.333718974\)
\(L(\frac12)\) \(\approx\) \(2.333718974\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 - 8.07i)T \)
good7 \( 1 + (0.388 - 0.673i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.02 - 3.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.61 + 2.79i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.72 - 2.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.784 - 1.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0608 + 0.105i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.58 + 7.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.20 + 5.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.07 - 3.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.36 - 9.28i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (2.40 - 4.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.61T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + (2.09 + 3.62i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (4.29 - 7.43i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.944 - 1.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.70 - 15.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.60 - 13.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.89T + 89T^{2} \)
97 \( 1 + (-1.00 - 1.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.271723729354244432258158087883, −8.028205523523675510574093383902, −7.23973961211208349502421039796, −6.28813585643391029718627618820, −5.65674611564237372958223795787, −4.73249677037487187425512889912, −4.02997796264784345782670433440, −2.77387252655767462422469370259, −2.41229500140401412418240623817, −1.15362218279467514890741357251, 0.66389680859088207033639702623, 1.92055847297052522815048529545, 2.89789777838704865545062606609, 3.46360346303514269539801840340, 4.63252359576339728226675441572, 5.27293864508235537143868207259, 6.14433031243878603902649562988, 7.07150625340309897110828562244, 7.44232037433386985309882254366, 8.632485047488986457897673765687

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