Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.817 - 0.575i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + (0.827 + 1.43i)7-s + 9-s + (−2.81 − 4.87i)11-s + (0.800 − 1.38i)13-s − 15-s + (−1.35 + 2.34i)17-s + (−0.454 + 0.787i)19-s + (−0.827 − 1.43i)21-s + (−3.02 + 5.23i)23-s + 25-s − 27-s + (−1.34 − 2.33i)29-s + (5.11 + 8.85i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (0.312 + 0.541i)7-s + 0.333·9-s + (−0.848 − 1.46i)11-s + (0.222 − 0.384i)13-s − 0.258·15-s + (−0.328 + 0.569i)17-s + (−0.104 + 0.180i)19-s + (−0.180 − 0.312i)21-s + (−0.630 + 1.09i)23-s + 0.200·25-s − 0.192·27-s + (−0.250 − 0.433i)29-s + (0.918 + 1.59i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.817 - 0.575i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (3781, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.817 - 0.575i)$
$L(1)$  $\approx$  $1.471914666$
$L(\frac12)$  $\approx$  $1.471914666$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
67 \( 1 + (-8.03 + 1.56i)T \)
good7 \( 1 + (-0.827 - 1.43i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.81 + 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.800 + 1.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.35 - 2.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.454 - 0.787i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.02 - 5.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.34 + 2.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.11 - 8.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.237 - 0.411i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.573 - 0.992i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.46T + 43T^{2} \)
47 \( 1 + (-1.69 - 2.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.97T + 53T^{2} \)
59 \( 1 - 8.25T + 59T^{2} \)
61 \( 1 + (-5.07 + 8.79i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-1.81 - 3.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.751 - 1.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.38 - 7.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.96 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + (-2.33 + 4.03i)T + (-48.5 - 84.0i)T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.245870066058211816054628339675, −8.129177275889083165596531638794, −6.89051105640831595605481999189, −6.10084234186512275327681164028, −5.57951792316504460638792741777, −5.08260823709505291736694019900, −3.89907380302460700497812162886, −3.02396411801247496114293711656, −2.03074699490986513008911496682, −0.869432823657595475603136525370, 0.58635853988780008527197485354, 1.93817508858286494727572929380, 2.58309619486762282626130739401, 4.14536025651149287436368292552, 4.54999930040703704331165008793, 5.34177557544895644643484831911, 6.19098278992304165711391627146, 6.94049506044008897202811779871, 7.50152284210648014166462221326, 8.257229603465600220375659625298

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