Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.0235 + 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.271 − 1.38i)2-s + 3-s + (−1.85 − 0.754i)4-s + 0.299i·5-s + (0.271 − 1.38i)6-s + 1.88·7-s + (−1.55 + 2.36i)8-s + 9-s + (0.416 + 0.0814i)10-s + 2.47·11-s + (−1.85 − 0.754i)12-s − 1.99i·13-s + (0.511 − 2.60i)14-s + 0.299i·15-s + (2.86 + 2.79i)16-s + 0.452·17-s + ⋯
L(s)  = 1  + (0.192 − 0.981i)2-s + 0.577·3-s + (−0.926 − 0.377i)4-s + 0.134i·5-s + (0.110 − 0.566i)6-s + 0.710·7-s + (−0.548 + 0.836i)8-s + 0.333·9-s + (0.131 + 0.0257i)10-s + 0.744·11-s + (−0.534 − 0.217i)12-s − 0.554i·13-s + (0.136 − 0.697i)14-s + 0.0774i·15-s + (0.715 + 0.698i)16-s + 0.109·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0235 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.0235 + 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.0235 + 0.999i)$
$L(1)$  $\approx$  $1.45669 - 1.49137i$
$L(\frac12)$  $\approx$  $1.45669 - 1.49137i$
$L(\frac{3}{2})$   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 \( 1 + (-0.271 + 1.38i)T \)
3 \( 1 - T \)
67 \( 1 + (2.90 + 7.65i)T \)
good5 \( 1 - 0.299iT - 5T^{2} \)
7 \( 1 - 1.88T + 7T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 + 1.99iT - 13T^{2} \)
17 \( 1 - 0.452T + 17T^{2} \)
19 \( 1 + 5.21iT - 19T^{2} \)
23 \( 1 + 0.807iT - 23T^{2} \)
29 \( 1 - 9.00T + 29T^{2} \)
31 \( 1 + 8.22T + 31T^{2} \)
37 \( 1 - 1.31T + 37T^{2} \)
41 \( 1 + 6.67iT - 41T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 - 2.51iT - 47T^{2} \)
53 \( 1 + 0.990iT - 53T^{2} \)
59 \( 1 - 7.56iT - 59T^{2} \)
61 \( 1 - 9.06iT - 61T^{2} \)
71 \( 1 - 0.0862iT - 71T^{2} \)
73 \( 1 - 1.91T + 73T^{2} \)
79 \( 1 + 8.30T + 79T^{2} \)
83 \( 1 - 0.827iT - 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 6.11iT - 97T^{2} \)
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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

−10.23023455926840830489382647467, −9.099209550757467471442733702516, −8.717215168833003384236030617062, −7.67796437476458184058706255739, −6.55931665797709763227224391495, −5.22629301399780743976567371741, −4.45431048405198661263026161603, −3.36655068944601587254822882515, −2.41197580599474770302676181716, −1.11514113643313398861056736268, 1.50925682611683350861064065841, 3.29159134021812515277685840432, 4.28023792157894009611131912807, 5.08419247557669900828117473125, 6.25126111278760367100692946359, 7.03163111623298162392637195088, 8.001313665584536329375111554535, 8.558294176528166512925063361229, 9.352198679078745397445448338765, 10.17611635523044158850484485223

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