Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.665 + 0.746i$
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 + (−1.48 + 1.66i)5-s − 2.27i·7-s − 9-s + 3.88·11-s − 2.72i·13-s + (1.66 + 1.48i)15-s − 3.85i·17-s + 1.23·19-s − 2.27·21-s − 0.161i·23-s + (−0.569 − 4.96i)25-s + i·27-s − 9.75·29-s + 10.8·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.665 + 0.746i)5-s − 0.861i·7-s − 0.333·9-s + 1.17·11-s − 0.754i·13-s + (0.430 + 0.384i)15-s − 0.934i·17-s + 0.283·19-s − 0.497·21-s − 0.0337i·23-s + (−0.113 − 0.993i)25-s + 0.192i·27-s − 1.81·29-s + 1.94·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.665 + 0.746i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.665 + 0.746i)$
$L(1)$  $\approx$  $1.203996138$
$L(\frac12)$  $\approx$  $1.203996138$
$L(\frac{3}{2})$   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 \( 1 + (1.48 - 1.66i)T \)
67 \( 1 + iT \)
good7 \( 1 + 2.27iT - 7T^{2} \)
11 \( 1 - 3.88T + 11T^{2} \)
13 \( 1 + 2.72iT - 13T^{2} \)
17 \( 1 + 3.85iT - 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 + 0.161iT - 23T^{2} \)
29 \( 1 + 9.75T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 - 4.23iT - 37T^{2} \)
41 \( 1 + 9.12T + 41T^{2} \)
43 \( 1 - 12.4iT - 43T^{2} \)
47 \( 1 + 7.41iT - 47T^{2} \)
53 \( 1 + 5.06iT - 53T^{2} \)
59 \( 1 + 0.0652T + 59T^{2} \)
61 \( 1 - 7.45T + 61T^{2} \)
71 \( 1 - 3.87T + 71T^{2} \)
73 \( 1 + 2.30iT - 73T^{2} \)
79 \( 1 + 8.08T + 79T^{2} \)
83 \( 1 + 8.86iT - 83T^{2} \)
89 \( 1 + 2.68T + 89T^{2} \)
97 \( 1 + 1.15iT - 97T^{2} \)
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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

−7.996085865289634429353643489383, −7.38677611007091125099481890122, −6.80192481647247192086609396880, −6.27914722846150675493916460759, −5.16947936703301928993385218055, −4.22230990446416239217050876693, −3.46514958711627672669192199661, −2.76182575158533446119291744870, −1.41828966001710017464711566441, −0.38047454202608651937071546207, 1.25501429881619787428823335275, 2.30322014621549856075226410957, 3.69032418684176798993957434175, 3.98780285965123950603422689492, 4.92766528178670508426813487245, 5.65525448215943421119143408827, 6.41410817688193719721919298033, 7.29334603654189877035417200670, 8.233418730467765297119523533797, 8.824962856457366544340659141423

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