Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.923 + 0.384i$
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 + (−2.06 + 0.858i)5-s + 3.58i·7-s − 9-s + 0.928·11-s + 5.62i·13-s + (0.858 + 2.06i)15-s + 6.87i·17-s − 7.74·19-s + 3.58·21-s + 6.77i·23-s + (3.52 − 3.54i)25-s + i·27-s − 2.69·29-s − 2.29·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.923 + 0.384i)5-s + 1.35i·7-s − 0.333·9-s + 0.279·11-s + 1.55i·13-s + (0.221 + 0.533i)15-s + 1.66i·17-s − 1.77·19-s + 0.782·21-s + 1.41i·23-s + (0.704 − 0.709i)25-s + 0.192i·27-s − 0.500·29-s − 0.412·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.923 + 0.384i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.923 + 0.384i)$
$L(1)$  $\approx$  $0.4304716417$
$L(\frac12)$  $\approx$  $0.4304716417$
$L(\frac{3}{2})$   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 \( 1 + (2.06 - 0.858i)T \)
67 \( 1 + iT \)
good7 \( 1 - 3.58iT - 7T^{2} \)
11 \( 1 - 0.928T + 11T^{2} \)
13 \( 1 - 5.62iT - 13T^{2} \)
17 \( 1 - 6.87iT - 17T^{2} \)
19 \( 1 + 7.74T + 19T^{2} \)
23 \( 1 - 6.77iT - 23T^{2} \)
29 \( 1 + 2.69T + 29T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 + 2.54T + 41T^{2} \)
43 \( 1 + 0.670iT - 43T^{2} \)
47 \( 1 + 10.2iT - 47T^{2} \)
53 \( 1 + 3.41iT - 53T^{2} \)
59 \( 1 - 1.49T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
71 \( 1 - 3.25T + 71T^{2} \)
73 \( 1 + 15.6iT - 73T^{2} \)
79 \( 1 + 4.76T + 79T^{2} \)
83 \( 1 - 6.48iT - 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−8.746659715629405401750397819079, −8.237889288938604024439318737972, −7.35053548543010217798727300315, −6.62467237253291583799021387679, −6.10249284157989422720963859045, −5.25684932611684557936977205701, −4.02474537372116272035317269965, −3.66986293238157542338050805392, −2.19549873976051500189201801461, −1.85737549517410565063177698792, 0.14754453505073799577999681073, 0.937506992051415636274535909603, 2.71017050905034828365821534961, 3.47257343715211311334143203012, 4.41182630498522888477648827464, 4.63140532521760138438972066409, 5.68211647383296005210981920112, 6.79334023379444939770558796033, 7.27378790658990661083684921887, 8.255860369573588343645790432110

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