Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.0566 - 0.998i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.351i)2-s − 3-s + (1.75 + 0.962i)4-s + 3.58i·5-s + (−1.36 − 0.351i)6-s + 4.07·7-s + (2.06 + 1.93i)8-s + 9-s + (−1.25 + 4.91i)10-s − 4.30·11-s + (−1.75 − 0.962i)12-s + 0.588i·13-s + (5.58 + 1.43i)14-s − 3.58i·15-s + (2.14 + 3.37i)16-s + 0.330·17-s + ⋯
L(s)  = 1  + (0.968 + 0.248i)2-s − 0.577·3-s + (0.876 + 0.481i)4-s + 1.60i·5-s + (−0.559 − 0.143i)6-s + 1.53·7-s + (0.729 + 0.683i)8-s + 0.333·9-s + (−0.398 + 1.55i)10-s − 1.29·11-s + (−0.506 − 0.277i)12-s + 0.163i·13-s + (1.49 + 0.382i)14-s − 0.926i·15-s + (0.537 + 0.843i)16-s + 0.0801·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0566 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0566 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.0566 - 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.0566 - 0.998i)$
$L(1)$  $\approx$  $1.78078 + 1.88477i$
$L(\frac12)$  $\approx$  $1.78078 + 1.88477i$
$L(\frac{3}{2})$   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 \( 1 + (-1.36 - 0.351i)T \)
3 \( 1 + T \)
67 \( 1 + (-4.33 - 6.94i)T \)
good5 \( 1 - 3.58iT - 5T^{2} \)
7 \( 1 - 4.07T + 7T^{2} \)
11 \( 1 + 4.30T + 11T^{2} \)
13 \( 1 - 0.588iT - 13T^{2} \)
17 \( 1 - 0.330T + 17T^{2} \)
19 \( 1 + 5.25iT - 19T^{2} \)
23 \( 1 - 0.501iT - 23T^{2} \)
29 \( 1 + 5.91T + 29T^{2} \)
31 \( 1 - 5.74T + 31T^{2} \)
37 \( 1 + 1.03T + 37T^{2} \)
41 \( 1 - 7.04iT - 41T^{2} \)
43 \( 1 - 6.86T + 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + 8.47iT - 53T^{2} \)
59 \( 1 - 8.93iT - 59T^{2} \)
61 \( 1 - 6.57iT - 61T^{2} \)
71 \( 1 + 0.0930iT - 71T^{2} \)
73 \( 1 + 6.32T + 73T^{2} \)
79 \( 1 + 1.43T + 79T^{2} \)
83 \( 1 + 12.5iT - 83T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + 6.83iT - 97T^{2} \)
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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

−10.77388905142531004156884110826, −10.16713515595551284483967187681, −8.398860926836578510906270992022, −7.47415678564901506962605462717, −7.07081052231366629625195367803, −5.96198234858967713836278773235, −5.17159709547168716087451535593, −4.33578686289813008562996591568, −2.97920631599108142835721022972, −2.09452040863470249340323657428, 1.08917398874196517166722722911, 2.12984622713311344348074902988, 3.97860652759347017685081705833, 4.88329697959021047579072576663, 5.24419741673969889647327984347, 6.02173729875720581962177385034, 7.75737237673896341226768217499, 7.951859919476437450210094231641, 9.263326353092201819738853660175, 10.41853702811164058765555243098

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