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  :  $\chi_{4020} (841, \cdot )$
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

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

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