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Properties

Label 2-4020-67.29-c1-0-34
Degree $2$
Conductor $4020$
Sign $0.108 + 0.994i$
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 + (−2.18 − 3.78i)7-s + 9-s + (1.49 + 2.59i)11-s + (−1.04 + 1.80i)13-s + 15-s + (−0.454 + 0.786i)17-s + (3.53 − 6.11i)19-s + (−2.18 − 3.78i)21-s + (3.49 − 6.04i)23-s + 25-s + 27-s + (3.93 + 6.82i)29-s + (−3.83 − 6.65i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (−0.826 − 1.43i)7-s + 0.333·9-s + (0.450 + 0.780i)11-s + (−0.288 + 0.500i)13-s + 0.258·15-s + (−0.110 + 0.190i)17-s + (0.810 − 1.40i)19-s + (−0.477 − 0.826i)21-s + (0.727 − 1.26i)23-s + 0.200·25-s + 0.192·27-s + (0.731 + 1.26i)29-s + (−0.689 − 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.108 + 0.994i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.108 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.152150734\)
\(L(\frac12)\) \(\approx\) \(2.152150734\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.50 + 7.79i)T \)
good7 \( 1 + (2.18 + 3.78i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.49 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.04 - 1.80i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.454 - 0.786i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.53 + 6.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.49 + 6.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.93 - 6.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.83 + 6.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.417 - 0.722i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.33 + 2.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + (-0.436 - 0.756i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.39T + 53T^{2} \)
59 \( 1 + 1.81T + 59T^{2} \)
61 \( 1 + (-1.99 + 3.44i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-0.0217 - 0.0377i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.49 + 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.453 + 0.785i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.41 + 2.44i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.76T + 89T^{2} \)
97 \( 1 + (-0.274 + 0.474i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.331131116130286482993456555991, −7.21348279614056248016105157197, −6.97915203911114138878017216160, −6.42977131548190960255483552347, −5.02394274178995325513401553689, −4.47037448616399456057842986679, −3.58905955231832658414730210144, −2.81772245999128301890967551670, −1.75032321520069711950298439218, −0.58485768703785367439546181915, 1.27558716054032251530298465531, 2.36695395109719928590992982189, 3.17071366567568691298612461218, 3.65709859391361156031792120217, 5.16966389647050672216152911671, 5.61233274721075178311783274932, 6.33419905895069952548049427081, 7.11581554101472096069553975846, 8.114544011882567381410246315698, 8.635878464772782579802216883379

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