Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.930 + 0.365i$
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 + (2.08 + 0.817i)5-s − 3.41i·7-s − 9-s + 5.18·11-s + 5.24i·13-s + (−0.817 + 2.08i)15-s − 7.93i·17-s + 1.59·19-s + 3.41·21-s − 7.16i·23-s + (3.66 + 3.40i)25-s i·27-s + 6.58·29-s − 5.96·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.930 + 0.365i)5-s − 1.29i·7-s − 0.333·9-s + 1.56·11-s + 1.45i·13-s + (−0.211 + 0.537i)15-s − 1.92i·17-s + 0.366·19-s + 0.745·21-s − 1.49i·23-s + (0.732 + 0.680i)25-s − 0.192i·27-s + 1.22·29-s − 1.07·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.930 + 0.365i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.930 + 0.365i)$
$L(1)$  $\approx$  $2.518755626$
$L(\frac12)$  $\approx$  $2.518755626$
$L(\frac{3}{2})$   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 \( 1 + (-2.08 - 0.817i)T \)
67 \( 1 - iT \)
good7 \( 1 + 3.41iT - 7T^{2} \)
11 \( 1 - 5.18T + 11T^{2} \)
13 \( 1 - 5.24iT - 13T^{2} \)
17 \( 1 + 7.93iT - 17T^{2} \)
19 \( 1 - 1.59T + 19T^{2} \)
23 \( 1 + 7.16iT - 23T^{2} \)
29 \( 1 - 6.58T + 29T^{2} \)
31 \( 1 + 5.96T + 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + 8.07T + 41T^{2} \)
43 \( 1 + 0.663iT - 43T^{2} \)
47 \( 1 + 7.08iT - 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 - 1.90T + 59T^{2} \)
61 \( 1 - 1.14T + 61T^{2} \)
71 \( 1 - 1.39T + 71T^{2} \)
73 \( 1 - 6.22iT - 73T^{2} \)
79 \( 1 + 1.75T + 79T^{2} \)
83 \( 1 + 12.9iT - 83T^{2} \)
89 \( 1 + 6.29T + 89T^{2} \)
97 \( 1 + 4.62iT - 97T^{2} \)
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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

−8.841169435813674535660120878114, −7.31691420792377514352596678890, −6.89691486590858108190036378572, −6.42579309766952533613948882155, −5.32468820391048675539681429558, −4.46273852854162946285812975762, −3.97689794254322665377606285285, −2.96319871169364041202832461111, −1.89722043444732609396296121935, −0.77531793749300556370000558209, 1.28457083334471427783624836864, 1.76903756596888634749497862436, 2.92447638493268212094984791043, 3.70733856996634512691974158178, 5.07178939822093742413695075149, 5.63342698728113647184777545737, 6.22656663412857461272197194958, 6.75414052367638760394817235908, 8.061386011959020146103357331019, 8.412780438064087503595071513133

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