Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.310 + 0.950i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 0.442i)2-s − 3-s + (1.60 + 1.18i)4-s − 2.27i·5-s + (−1.34 − 0.442i)6-s − 4.97·7-s + (1.63 + 2.30i)8-s + 9-s + (1.00 − 3.05i)10-s − 2.26·11-s + (−1.60 − 1.18i)12-s − 6.26i·13-s + (−6.68 − 2.19i)14-s + 2.27i·15-s + (1.17 + 3.82i)16-s − 0.801·17-s + ⋯
L(s)  = 1  + (0.949 + 0.312i)2-s − 0.577·3-s + (0.804 + 0.593i)4-s − 1.01i·5-s + (−0.548 − 0.180i)6-s − 1.88·7-s + (0.578 + 0.815i)8-s + 0.333·9-s + (0.317 − 0.964i)10-s − 0.681·11-s + (−0.464 − 0.342i)12-s − 1.73i·13-s + (−1.78 − 0.587i)14-s + 0.586i·15-s + (0.294 + 0.955i)16-s − 0.194·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.310 + 0.950i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.310 + 0.950i)$
$L(1)$  $\approx$  $0.641026 - 0.884038i$
$L(\frac12)$  $\approx$  $0.641026 - 0.884038i$
$L(\frac{3}{2})$   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 \( 1 + (-1.34 - 0.442i)T \)
3 \( 1 + T \)
67 \( 1 + (2.57 + 7.77i)T \)
good5 \( 1 + 2.27iT - 5T^{2} \)
7 \( 1 + 4.97T + 7T^{2} \)
11 \( 1 + 2.26T + 11T^{2} \)
13 \( 1 + 6.26iT - 13T^{2} \)
17 \( 1 + 0.801T + 17T^{2} \)
19 \( 1 + 2.05iT - 19T^{2} \)
23 \( 1 + 7.41iT - 23T^{2} \)
29 \( 1 + 6.30T + 29T^{2} \)
31 \( 1 - 2.55T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + 0.0427iT - 41T^{2} \)
43 \( 1 - 0.122T + 43T^{2} \)
47 \( 1 - 4.63iT - 47T^{2} \)
53 \( 1 + 6.54iT - 53T^{2} \)
59 \( 1 - 4.77iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
71 \( 1 + 4.26iT - 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 1.63T + 79T^{2} \)
83 \( 1 + 5.62iT - 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 14.5iT - 97T^{2} \)
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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

−10.29817090735073282354567325811, −9.115469304096980608453179000111, −8.169532946363812164389808065682, −7.12817774139519117846969351256, −6.28601641336371651747015522691, −5.52705874031497066416610197556, −4.82244448134597527563410838478, −3.58619488609435977279060081976, −2.66802054512449387906177496605, −0.39159835404168123447899487524, 2.05562233641326792885421349448, 3.27440453299787642947007439797, 3.88359140159627655290549946572, 5.29322838895890345437915954899, 6.19659976688907755340744224819, 6.79918904931740101535280618050, 7.31790319068582852576972186078, 9.276609034340949940249262066695, 9.914644093530665761745350025944, 10.63405165362110205040846048949

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