Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.990 + 0.141i$
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 + (2.21 + 0.315i)5-s − 0.0918i·7-s − 9-s − 3.06·11-s + 5.35i·13-s + (0.315 − 2.21i)15-s − 2.02i·17-s + 7.88·19-s − 0.0918·21-s − 5.50i·23-s + (4.80 + 1.39i)25-s + i·27-s − 2.34·29-s − 1.86·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.990 + 0.141i)5-s − 0.0347i·7-s − 0.333·9-s − 0.923·11-s + 1.48i·13-s + (0.0814 − 0.571i)15-s − 0.492i·17-s + 1.80·19-s − 0.0200·21-s − 1.14i·23-s + (0.960 + 0.279i)25-s + 0.192i·27-s − 0.435·29-s − 0.335·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.990 + 0.141i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.990 + 0.141i)$
$L(1)$  $\approx$  $2.245970564$
$L(\frac12)$  $\approx$  $2.245970564$
$L(\frac{3}{2})$   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 \( 1 + (-2.21 - 0.315i)T \)
67 \( 1 + iT \)
good7 \( 1 + 0.0918iT - 7T^{2} \)
11 \( 1 + 3.06T + 11T^{2} \)
13 \( 1 - 5.35iT - 13T^{2} \)
17 \( 1 + 2.02iT - 17T^{2} \)
19 \( 1 - 7.88T + 19T^{2} \)
23 \( 1 + 5.50iT - 23T^{2} \)
29 \( 1 + 2.34T + 29T^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 - 6.88iT - 37T^{2} \)
41 \( 1 + 5.27T + 41T^{2} \)
43 \( 1 + 6.19iT - 43T^{2} \)
47 \( 1 - 3.51iT - 47T^{2} \)
53 \( 1 + 1.50iT - 53T^{2} \)
59 \( 1 - 7.85T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 - 6.48iT - 73T^{2} \)
79 \( 1 - 8.72T + 79T^{2} \)
83 \( 1 - 3.43iT - 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 + 3.65iT - 97T^{2} \)
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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

−8.482642456440773516194242393028, −7.53912559928598752586365227139, −6.91063765796460228030642386795, −6.38569337090952876398296511585, −5.33846105950175498337387092027, −5.01676815549585183268366425966, −3.70685672979882029831063929323, −2.64433019345373162613132955043, −2.04770441183105429666464355002, −0.930621101673477384771255784771, 0.812959965645074319168758609190, 2.09670001557287879230279538323, 3.05054759036481549937507292321, 3.68378498917271482507744842828, 5.08226337726605185505685732733, 5.42326641902881677285774995897, 5.86588320455952925695819276656, 7.09954530035153848008023098724, 7.79537312366796050103623129187, 8.467201346295137872455597357499

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