Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.312 + 0.949i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.67i·2-s − 1.67·3-s − 5.15·4-s + 2.86i·5-s − 4.46i·6-s + 5.21i·7-s − 8.44i·8-s − 0.209·9-s − 7.66·10-s − 0.107i·11-s + 8.61·12-s + (3.42 − 1.12i)13-s − 13.9·14-s − 4.78i·15-s + 12.2·16-s − 1.65·17-s + ⋯
L(s)  = 1  + 1.89i·2-s − 0.964·3-s − 2.57·4-s + 1.28i·5-s − 1.82i·6-s + 1.96i·7-s − 2.98i·8-s − 0.0698·9-s − 2.42·10-s − 0.0323i·11-s + 2.48·12-s + (0.949 − 0.312i)13-s − 3.72·14-s − 1.23i·15-s + 3.06·16-s − 0.401·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.312 + 0.949i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.312 + 0.949i)$
$L(1)$  $\approx$  $0.514065 - 0.372139i$
$L(\frac12)$  $\approx$  $0.514065 - 0.372139i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{13,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (-3.42 + 1.12i)T \)
31 \( 1 + iT \)
good2 \( 1 - 2.67iT - 2T^{2} \)
3 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 - 2.86iT - 5T^{2} \)
7 \( 1 - 5.21iT - 7T^{2} \)
11 \( 1 + 0.107iT - 11T^{2} \)
17 \( 1 + 1.65T + 17T^{2} \)
19 \( 1 - 2.74iT - 19T^{2} \)
23 \( 1 - 2.05T + 23T^{2} \)
29 \( 1 - 8.54T + 29T^{2} \)
37 \( 1 + 7.15iT - 37T^{2} \)
41 \( 1 - 9.28iT - 41T^{2} \)
43 \( 1 + 7.32T + 43T^{2} \)
47 \( 1 + 4.70iT - 47T^{2} \)
53 \( 1 + 3.04T + 53T^{2} \)
59 \( 1 - 11.4iT - 59T^{2} \)
61 \( 1 + 5.46T + 61T^{2} \)
67 \( 1 - 1.60iT - 67T^{2} \)
71 \( 1 - 8.50iT - 71T^{2} \)
73 \( 1 + 6.98iT - 73T^{2} \)
79 \( 1 + 1.90T + 79T^{2} \)
83 \( 1 - 9.64iT - 83T^{2} \)
89 \( 1 + 9.70iT - 89T^{2} \)
97 \( 1 + 14.1iT - 97T^{2} \)
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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

−12.00835959945236987752287788093, −11.10334853868626082876066501791, −9.992554381831783206227288066930, −8.792450475129345069441919249709, −8.279550299972380336111718512902, −6.94254678416154925795155561372, −6.08564227730464750959918551555, −5.89814862460506081247698181508, −4.83217454028895991535604064226, −3.06189212499361988760764239449, 0.54772789827394013181865708734, 1.31477744630168698078845831534, 3.41636591831270242939291857552, 4.55975783314967675625157309752, 4.90227262001598199621769166222, 6.57022403713356969113180975783, 8.228864947728068757936702329040, 9.001925050725667133361829066979, 10.07938931054557473299832694617, 10.77314546453488965655898301064

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