Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.877 - 0.479i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.44i·2-s − 2.79·3-s − 0.0740·4-s + 2.61i·5-s + 4.02i·6-s − 4.99i·7-s − 2.77i·8-s + 4.79·9-s + 3.76·10-s − 0.0546i·11-s + 0.206·12-s + (−1.72 + 3.16i)13-s − 7.19·14-s − 7.30i·15-s − 4.14·16-s − 5.08·17-s + ⋯
L(s)  = 1  − 1.01i·2-s − 1.61·3-s − 0.0370·4-s + 1.17i·5-s + 1.64i·6-s − 1.88i·7-s − 0.980i·8-s + 1.59·9-s + 1.19·10-s − 0.0164i·11-s + 0.0596·12-s + (−0.479 + 0.877i)13-s − 1.92·14-s − 1.88i·15-s − 1.03·16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.877 - 0.479i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ -0.877 - 0.479i)$
$L(1)$  $\approx$  $0.0864821 + 0.338640i$
$L(\frac12)$  $\approx$  $0.0864821 + 0.338640i$
$L(\frac{3}{2})$   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 \( 1 + (1.72 - 3.16i)T \)
31 \( 1 + iT \)
good2 \( 1 + 1.44iT - 2T^{2} \)
3 \( 1 + 2.79T + 3T^{2} \)
5 \( 1 - 2.61iT - 5T^{2} \)
7 \( 1 + 4.99iT - 7T^{2} \)
11 \( 1 + 0.0546iT - 11T^{2} \)
17 \( 1 + 5.08T + 17T^{2} \)
19 \( 1 + 1.17iT - 19T^{2} \)
23 \( 1 + 7.81T + 23T^{2} \)
29 \( 1 + 7.02T + 29T^{2} \)
37 \( 1 + 6.75iT - 37T^{2} \)
41 \( 1 - 1.55iT - 41T^{2} \)
43 \( 1 + 3.02T + 43T^{2} \)
47 \( 1 + 0.191iT - 47T^{2} \)
53 \( 1 - 8.59T + 53T^{2} \)
59 \( 1 - 2.91iT - 59T^{2} \)
61 \( 1 - 5.96T + 61T^{2} \)
67 \( 1 + 1.50iT - 67T^{2} \)
71 \( 1 + 1.44iT - 71T^{2} \)
73 \( 1 + 13.9iT - 73T^{2} \)
79 \( 1 - 1.10T + 79T^{2} \)
83 \( 1 + 1.74iT - 83T^{2} \)
89 \( 1 + 2.35iT - 89T^{2} \)
97 \( 1 - 12.1iT - 97T^{2} \)
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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

−10.91502583286785697815116286712, −10.36167610554490191232736084698, −9.646220093081246521847593216088, −7.33616015589028755212547343394, −6.93125626365828164509234983284, −6.18224049505008884362915855013, −4.45710706867402780018642835265, −3.77354268919367901386869203468, −1.98158137384099356205838194186, −0.25804067569286527232360389081, 2.10827090668192227140480631913, 4.68583247118307069231671613724, 5.50785273667435810174292331403, 5.77571498824163196943530938129, 6.76260828920159174551202652673, 8.115301402006030270615642845753, 8.777156606064778261000565812320, 9.911362876049591347056534383065, 11.21252878743714168284381573120, 11.87258707217521056970185045340

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