Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.977 + 0.209i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.71i·2-s + 0.265·3-s − 5.36·4-s − 0.954i·5-s + 0.721i·6-s − 4.28i·7-s − 9.13i·8-s − 2.92·9-s + 2.58·10-s − 0.0225i·11-s − 1.42·12-s + (0.755 − 3.52i)13-s + 11.6·14-s − 0.253i·15-s + 14.0·16-s − 4.73·17-s + ⋯
L(s)  = 1  + 1.91i·2-s + 0.153·3-s − 2.68·4-s − 0.426i·5-s + 0.294i·6-s − 1.61i·7-s − 3.22i·8-s − 0.976·9-s + 0.818·10-s − 0.00678i·11-s − 0.411·12-s + (0.209 − 0.977i)13-s + 3.10·14-s − 0.0654i·15-s + 3.51·16-s − 1.14·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.977 + 0.209i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.977 + 0.209i)$
$L(1)$  $\approx$  $0.653440 - 0.0692657i$
$L(\frac12)$  $\approx$  $0.653440 - 0.0692657i$
$L(\frac{3}{2})$   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 \( 1 + (-0.755 + 3.52i)T \)
31 \( 1 + iT \)
good2 \( 1 - 2.71iT - 2T^{2} \)
3 \( 1 - 0.265T + 3T^{2} \)
5 \( 1 + 0.954iT - 5T^{2} \)
7 \( 1 + 4.28iT - 7T^{2} \)
11 \( 1 + 0.0225iT - 11T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 - 1.51iT - 19T^{2} \)
23 \( 1 + 7.63T + 23T^{2} \)
29 \( 1 + 2.41T + 29T^{2} \)
37 \( 1 - 2.46iT - 37T^{2} \)
41 \( 1 + 6.00iT - 41T^{2} \)
43 \( 1 - 9.18T + 43T^{2} \)
47 \( 1 - 6.05iT - 47T^{2} \)
53 \( 1 + 5.67T + 53T^{2} \)
59 \( 1 + 8.26iT - 59T^{2} \)
61 \( 1 - 9.28T + 61T^{2} \)
67 \( 1 + 7.52iT - 67T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + 1.76iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 13.3iT - 83T^{2} \)
89 \( 1 + 6.66iT - 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−10.97948357577349487255105690701, −10.01682378628377072765707694943, −8.985865351683217631060460951519, −8.147965551622899697425297600728, −7.58498493506782658899945236248, −6.54064012908674438437497814498, −5.69224525312666656844107659800, −4.59342993213682063687058467321, −3.70528574463233296675654393930, −0.41489555450784126834565589529, 2.12962625862493964227764309698, 2.66480443416022957560624528638, 3.94112232960077448631019972841, 5.16396320088319407315040491361, 6.23844685099547457350647770252, 8.258935562039015828478262271708, 8.980549374664927104454213063919, 9.424755182912250136311848689395, 10.69663411392135735976762705023, 11.42096909230972130329776978513

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