Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.268 - 0.963i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + (−0.599 − 2.15i)5-s + 0.0778i·7-s − 9-s − 4.03·11-s − 0.176i·13-s + (−2.15 + 0.599i)15-s − 7.72i·17-s − 1.88·19-s + 0.0778·21-s − 2.47i·23-s + (−4.28 + 2.58i)25-s + i·27-s + 6.84·29-s − 7.45·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.268 − 0.963i)5-s + 0.0294i·7-s − 0.333·9-s − 1.21·11-s − 0.0488i·13-s + (−0.556 + 0.154i)15-s − 1.87i·17-s − 0.432·19-s + 0.0169·21-s − 0.515i·23-s + (−0.856 + 0.516i)25-s + 0.192i·27-s + 1.27·29-s − 1.33·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.268 - 0.963i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.268 - 0.963i)$
$L(1)$  $\approx$  $0.1859373197$
$L(\frac12)$  $\approx$  $0.1859373197$
$L(\frac{3}{2})$   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 \( 1 + iT \)
5 \( 1 + (0.599 + 2.15i)T \)
67 \( 1 + iT \)
good7 \( 1 - 0.0778iT - 7T^{2} \)
11 \( 1 + 4.03T + 11T^{2} \)
13 \( 1 + 0.176iT - 13T^{2} \)
17 \( 1 + 7.72iT - 17T^{2} \)
19 \( 1 + 1.88T + 19T^{2} \)
23 \( 1 + 2.47iT - 23T^{2} \)
29 \( 1 - 6.84T + 29T^{2} \)
31 \( 1 + 7.45T + 31T^{2} \)
37 \( 1 + 3.67iT - 37T^{2} \)
41 \( 1 + 0.0931T + 41T^{2} \)
43 \( 1 - 6.63iT - 43T^{2} \)
47 \( 1 - 8.43iT - 47T^{2} \)
53 \( 1 + 1.60iT - 53T^{2} \)
59 \( 1 + 0.795T + 59T^{2} \)
61 \( 1 + 5.94T + 61T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 0.963iT - 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 5.08iT - 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−7.80892880040491032938278641609, −7.43250371310162499778012081650, −6.51126071677289583710164307996, −5.54138597114563527158638080685, −5.00661013722827384929203201184, −4.28131257198626721730702997246, −3.00529903325005597684213066868, −2.31105139330114217656546758676, −1.02069520164828616287942321101, −0.05838486879599218457472752514, 1.87238307920115725089148761476, 2.79525029800684158215477297810, 3.62861251164569507597899494318, 4.25640750808090075544942820042, 5.32867042811146742292087193282, 5.94833346776505449470519230839, 6.77212118069320989875177049413, 7.54541700204129545532988444004, 8.267499916001001351691692659505, 8.796992484009028702087313382832

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