Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.775 + 0.631i$
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 + (−0.952 − 1.64i)7-s + 9-s + (−1.50 − 2.60i)11-s + (2.70 − 4.69i)13-s − 15-s + (−2.26 + 3.92i)17-s + (1.93 − 3.35i)19-s + (0.952 + 1.64i)21-s + (1.94 − 3.36i)23-s + 25-s − 27-s + (4.12 + 7.13i)29-s + (−3.48 − 6.04i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (−0.359 − 0.623i)7-s + 0.333·9-s + (−0.453 − 0.784i)11-s + (0.751 − 1.30i)13-s − 0.258·15-s + (−0.550 + 0.953i)17-s + (0.444 − 0.770i)19-s + (0.207 + 0.359i)21-s + (0.405 − 0.702i)23-s + 0.200·25-s − 0.192·27-s + (0.765 + 1.32i)29-s + (−0.626 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.775 + 0.631i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (3781, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.775 + 0.631i)$
$L(1)$  $\approx$  $1.037146550$
$L(\frac12)$  $\approx$  $1.037146550$
$L(\frac{3}{2})$   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 \( 1 + (7.90 - 2.13i)T \)
good7 \( 1 + (0.952 + 1.64i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.50 + 2.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.70 + 4.69i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.26 - 3.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.93 + 3.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.94 + 3.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.12 - 7.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.48 + 6.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.725 + 1.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0559 - 0.0969i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.08T + 43T^{2} \)
47 \( 1 + (0.778 + 1.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.39T + 53T^{2} \)
59 \( 1 + 2.69T + 59T^{2} \)
61 \( 1 + (0.711 - 1.23i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-2.11 - 3.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.91 - 5.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.12 - 5.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.214 + 0.371i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.70T + 89T^{2} \)
97 \( 1 + (1.52 - 2.64i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.244097363570642929943073696568, −7.31210212812475883525658595111, −6.59690081342111346620177795892, −5.89570103554231261935138977708, −5.35691017678377365568823635608, −4.41252972094219940238591545112, −3.46074091440220574462203528076, −2.70886955605089057497339544066, −1.28196914077881624124264366203, −0.34316976677185336018798266741, 1.37271409756193811370178245979, 2.25073307594136352199245820575, 3.26044502574477670403091610555, 4.39546854916104016221984679476, 4.99549690832412979816398698352, 5.87383227668684156810134027397, 6.42774296258141041782083647004, 7.13396170205573160318585458997, 7.907007764333269497537491632792, 9.009168587219723107087417319847

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