Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.512 - 0.858i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s − 3-s + (−0.997 − 1.73i)4-s − 3.14i·5-s + (−0.707 + 1.22i)6-s − 1.38·7-s + (−2.82 − 0.00564i)8-s + 9-s + (−3.85 − 2.22i)10-s − 2.19·11-s + (0.997 + 1.73i)12-s − 1.11i·13-s + (−0.978 + 1.69i)14-s + 3.14i·15-s + (−2.00 + 3.45i)16-s + 0.639·17-s + ⋯
L(s)  = 1  + (0.500 − 0.865i)2-s − 0.577·3-s + (−0.498 − 0.866i)4-s − 1.40i·5-s + (−0.289 + 0.499i)6-s − 0.522·7-s + (−0.999 − 0.00199i)8-s + 0.333·9-s + (−1.21 − 0.704i)10-s − 0.660·11-s + (0.288 + 0.500i)12-s − 0.308i·13-s + (−0.261 + 0.452i)14-s + 0.812i·15-s + (−0.502 + 0.864i)16-s + 0.155·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.512 - 0.858i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.512 - 0.858i)$
$L(1)$  $\approx$  $0.341290 + 0.601102i$
$L(\frac12)$  $\approx$  $0.341290 + 0.601102i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 + T \)
67 \( 1 + (8.18 - 0.128i)T \)
good5 \( 1 + 3.14iT - 5T^{2} \)
7 \( 1 + 1.38T + 7T^{2} \)
11 \( 1 + 2.19T + 11T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 - 0.639T + 17T^{2} \)
19 \( 1 + 2.14iT - 19T^{2} \)
23 \( 1 - 5.50iT - 23T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + 4.75T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 2.53iT - 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 + 2.88iT - 47T^{2} \)
53 \( 1 + 0.00678iT - 53T^{2} \)
59 \( 1 - 0.119iT - 59T^{2} \)
61 \( 1 + 10.3iT - 61T^{2} \)
71 \( 1 - 5.27iT - 71T^{2} \)
73 \( 1 - 2.16T + 73T^{2} \)
79 \( 1 + 1.72T + 79T^{2} \)
83 \( 1 - 6.31iT - 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 5.12iT - 97T^{2} \)
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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

−9.818788137325437253994058104461, −9.103170485924370133965611228089, −8.242457949990836280765528594574, −6.91041077686665724565917359823, −5.62782967322416790563373947012, −5.22202575072954282495675132673, −4.29816573265623396562999887013, −3.14149455281380118669984480212, −1.60020110642840751213879403323, −0.31042252254433395966927619500, 2.63313747698435926602930172460, 3.54806606740614688263471575630, 4.69950281861856256729878432420, 5.78961353806703108746696003462, 6.51244655179109106634904964258, 7.07324902526439206802835260513, 7.934354177918324703826740441748, 9.029724851324716050334442317400, 10.21327424442332215683104944008, 10.65980331361824544349791354625

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