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Properties

Label 2-4020-201.200-c1-0-60
Degree $2$
Conductor $4020$
Sign $0.709 + 0.704i$
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  + (0.963 − 1.43i)3-s + 5-s + 2.54i·7-s + (−1.14 − 2.77i)9-s + 4.55·11-s + 2.49i·13-s + (0.963 − 1.43i)15-s − 7.63i·17-s + 2.99·19-s + (3.66 + 2.45i)21-s − 4.97i·23-s + 25-s + (−5.09 − 1.02i)27-s + 9.54i·29-s + 4.41i·31-s + ⋯
L(s)  = 1  + (0.556 − 0.831i)3-s + 0.447·5-s + 0.961i·7-s + (−0.381 − 0.924i)9-s + 1.37·11-s + 0.692i·13-s + (0.248 − 0.371i)15-s − 1.85i·17-s + 0.686·19-s + (0.799 + 0.534i)21-s − 1.03i·23-s + 0.200·25-s + (−0.980 − 0.197i)27-s + 1.77i·29-s + 0.793i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.874241040\)
\(L(\frac12)\) \(\approx\) \(2.874241040\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.963 + 1.43i)T \)
5 \( 1 - T \)
67 \( 1 + (-1.56 + 8.03i)T \)
good7 \( 1 - 2.54iT - 7T^{2} \)
11 \( 1 - 4.55T + 11T^{2} \)
13 \( 1 - 2.49iT - 13T^{2} \)
17 \( 1 + 7.63iT - 17T^{2} \)
19 \( 1 - 2.99T + 19T^{2} \)
23 \( 1 + 4.97iT - 23T^{2} \)
29 \( 1 - 9.54iT - 29T^{2} \)
31 \( 1 - 4.41iT - 31T^{2} \)
37 \( 1 + 1.29T + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 8.78iT - 43T^{2} \)
47 \( 1 - 2.76iT - 47T^{2} \)
53 \( 1 - 8.39T + 53T^{2} \)
59 \( 1 - 0.920iT - 59T^{2} \)
61 \( 1 - 4.62iT - 61T^{2} \)
71 \( 1 + 0.0757iT - 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 - 1.43iT - 79T^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 - 0.638iT - 89T^{2} \)
97 \( 1 - 6.99iT - 97T^{2} \)
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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

−8.785212813697106929104114853444, −7.35258589939274851567336641827, −7.00349186202457129120056448281, −6.30113343215481325493003611946, −5.49505548348053512801407263567, −4.64949427613836870775404595316, −3.46191723981255313950965187039, −2.71297440247808080851348389933, −1.89158082084352380677252662004, −0.928001654811334893039870428288, 1.08696754379833244906145282433, 2.11781867310981740184026324436, 3.34562848222196820268032937783, 3.94417772669580396075130572284, 4.45857770097981023778084134032, 5.70476562631361946367249827207, 6.10725256205404114741476076266, 7.23658899526002799116858087680, 7.893536006749274629923537600913, 8.564361639433880970004903485600

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