Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.756 + 0.653i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + (1.07 + 1.86i)7-s + 9-s + (2.31 + 4.00i)11-s + (2.99 − 5.19i)13-s − 15-s + (3.06 − 5.31i)17-s + (2.62 − 4.54i)19-s + (−1.07 − 1.86i)21-s + (−0.436 + 0.756i)23-s + 25-s − 27-s + (−2.60 − 4.50i)29-s + (1.56 + 2.71i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (0.407 + 0.705i)7-s + 0.333·9-s + (0.697 + 1.20i)11-s + (0.831 − 1.44i)13-s − 0.258·15-s + (0.743 − 1.28i)17-s + (0.601 − 1.04i)19-s + (−0.235 − 0.407i)21-s + (−0.0910 + 0.157i)23-s + 0.200·25-s − 0.192·27-s + (−0.483 − 0.837i)29-s + (0.281 + 0.488i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.756 + 0.653i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (3781, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.756 + 0.653i)$
$L(1)$  $\approx$  $1.962269910$
$L(\frac12)$  $\approx$  $1.962269910$
$L(\frac{3}{2})$   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 \( 1 + T \)
5 \( 1 - T \)
67 \( 1 + (-3.47 - 7.41i)T \)
good7 \( 1 + (-1.07 - 1.86i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.31 - 4.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.99 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.06 + 5.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.62 + 4.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.436 - 0.756i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.60 + 4.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.56 - 2.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.89 + 3.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.79 + 8.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.46T + 43T^{2} \)
47 \( 1 + (3.90 + 6.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.77T + 53T^{2} \)
59 \( 1 - 5.61T + 59T^{2} \)
61 \( 1 + (1.75 - 3.03i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-2.58 - 4.48i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.841 - 1.45i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.70 + 4.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.83 + 13.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.97T + 89T^{2} \)
97 \( 1 + (-5.01 + 8.68i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.415070758139577926737612102300, −7.46848176632018880203381932760, −6.94800360022692747333258308407, −6.01730262778957795213661418936, −5.26991382009119725256481417647, −4.96975601016880334998068842798, −3.74894920511816633912188637560, −2.77523677596611259481750597491, −1.77888440389977831971128789591, −0.68410962984173724828145689778, 1.26604483214766713595862910451, 1.56155096537628912344190344109, 3.35888878014467608075248048856, 3.89281546289267403468870826280, 4.75977250267261982160647398789, 5.77054936541997273453720293279, 6.28396751377137535370372980475, 6.78949692761478991526927127462, 7.984144919234325824828164233739, 8.345901909797083754156807745083

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