Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.996 - 0.0877i$
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 + (−2.22 − 0.196i)5-s − 0.128i·7-s − 9-s + 0.0389·11-s + 3.81i·13-s + (0.196 − 2.22i)15-s − 6.63i·17-s + 7.30·19-s + 0.128·21-s + 3.64i·23-s + (4.92 + 0.874i)25-s i·27-s − 8.25·29-s − 7.07·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.996 − 0.0877i)5-s − 0.0484i·7-s − 0.333·9-s + 0.0117·11-s + 1.05i·13-s + (0.0506 − 0.575i)15-s − 1.61i·17-s + 1.67·19-s + 0.0279·21-s + 0.760i·23-s + (0.984 + 0.174i)25-s − 0.192i·27-s − 1.53·29-s − 1.27·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0877i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0877i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.996 - 0.0877i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.996 - 0.0877i)$
$L(1)$  $\approx$  $0.4818936313$
$L(\frac12)$  $\approx$  $0.4818936313$
$L(\frac{3}{2})$   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 \( 1 + (2.22 + 0.196i)T \)
67 \( 1 - iT \)
good7 \( 1 + 0.128iT - 7T^{2} \)
11 \( 1 - 0.0389T + 11T^{2} \)
13 \( 1 - 3.81iT - 13T^{2} \)
17 \( 1 + 6.63iT - 17T^{2} \)
19 \( 1 - 7.30T + 19T^{2} \)
23 \( 1 - 3.64iT - 23T^{2} \)
29 \( 1 + 8.25T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 - 7.37iT - 37T^{2} \)
41 \( 1 - 1.87T + 41T^{2} \)
43 \( 1 - 7.72iT - 43T^{2} \)
47 \( 1 + 9.62iT - 47T^{2} \)
53 \( 1 - 3.82iT - 53T^{2} \)
59 \( 1 - 9.48T + 59T^{2} \)
61 \( 1 + 6.69T + 61T^{2} \)
71 \( 1 - 4.88T + 71T^{2} \)
73 \( 1 - 0.309iT - 73T^{2} \)
79 \( 1 + 5.72T + 79T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 7.10iT - 97T^{2} \)
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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

−9.028986782554265495047851790792, −7.980967179520593041333295113398, −7.29809592936895279048971720150, −6.91217158868810171671435778468, −5.52975331715437766184696760643, −5.09053003920086898807316188886, −4.15213626918606380001585070050, −3.54425498816033074318323736309, −2.70698768380421271903567466828, −1.26047582164221069277939420141, 0.15415128468608667053240376003, 1.32183677331753648790825581456, 2.52302740711017787918785386685, 3.53728295655674798426571014404, 3.99986485153594544459537675384, 5.34272797929525699975086512922, 5.74210188381963398742446657693, 6.79456727860522899945176441028, 7.61188734425633525235756564573, 7.79349100138725302024660829997

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