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Properties

Label 2-4020-67.37-c1-0-26
Degree $2$
Conductor $4020$
Sign $0.978 + 0.205i$
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 + (−0.5 + 0.866i)7-s + 9-s + (2.5 − 4.33i)11-s + (0.5 + 0.866i)13-s − 15-s + (3.5 + 6.06i)17-s + (2.5 + 4.33i)19-s + (−0.5 + 0.866i)21-s + (−4.5 − 7.79i)23-s + 25-s + 27-s + (1.5 − 2.59i)29-s + (3.5 − 6.06i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (−0.188 + 0.327i)7-s + 0.333·9-s + (0.753 − 1.30i)11-s + (0.138 + 0.240i)13-s − 0.258·15-s + (0.848 + 1.47i)17-s + (0.573 + 0.993i)19-s + (−0.109 + 0.188i)21-s + (−0.938 − 1.62i)23-s + 0.200·25-s + 0.192·27-s + (0.278 − 0.482i)29-s + (0.628 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.978 + 0.205i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.978 + 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.313195828\)
\(L(\frac12)\) \(\approx\) \(2.313195828\)
\(L(\frac{3}{2})\) 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 \( 1 + (-8 + 1.73i)T \)
good7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (3.5 - 6.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.5 + 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.382831543590931527937299037393, −7.999913497297040887747273978302, −6.98794517650558388553135528434, −5.98319911167567829114823617389, −5.83617337713193757661046648480, −4.26961363206489135727693717730, −3.83210798007507995258168335328, −3.06242961692403557541837558041, −1.97904142124449571270731278573, −0.816133910723787335645094288906, 0.936755714359830478958423710790, 2.00290762937205632472980029859, 3.23087212954232791513214625792, 3.61604525469188539450665387974, 4.80835440488526404172458958379, 5.16471241346136391153361672735, 6.61439409289064639552061258734, 7.10408990847386886559304565138, 7.62008891986542317429092759355, 8.449243014539680520881943608245

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