Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.0151 + 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.327i·2-s − 1.21·3-s + 1.89·4-s − 2.01i·5-s + 0.396i·6-s + 0.699i·7-s − 1.27i·8-s − 1.53·9-s − 0.660·10-s − 4.61i·11-s − 2.29·12-s + (3.60 + 0.0546i)13-s + 0.228·14-s + 2.44i·15-s + 3.36·16-s − 4.66·17-s + ⋯
L(s)  = 1  − 0.231i·2-s − 0.699·3-s + 0.946·4-s − 0.902i·5-s + 0.161i·6-s + 0.264i·7-s − 0.450i·8-s − 0.510·9-s − 0.208·10-s − 1.39i·11-s − 0.662·12-s + (0.999 + 0.0151i)13-s + 0.0611·14-s + 0.631i·15-s + 0.842·16-s − 1.13·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0151 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0151 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.0151 + 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ -0.0151 + 0.999i)$
$L(1)$  $\approx$  $0.861501 - 0.874656i$
$L(\frac12)$  $\approx$  $0.861501 - 0.874656i$
$L(\frac{3}{2})$   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 \( 1 + (-3.60 - 0.0546i)T \)
31 \( 1 + iT \)
good2 \( 1 + 0.327iT - 2T^{2} \)
3 \( 1 + 1.21T + 3T^{2} \)
5 \( 1 + 2.01iT - 5T^{2} \)
7 \( 1 - 0.699iT - 7T^{2} \)
11 \( 1 + 4.61iT - 11T^{2} \)
17 \( 1 + 4.66T + 17T^{2} \)
19 \( 1 + 2.12iT - 19T^{2} \)
23 \( 1 + 0.894T + 23T^{2} \)
29 \( 1 + 3.00T + 29T^{2} \)
37 \( 1 + 9.14iT - 37T^{2} \)
41 \( 1 + 8.88iT - 41T^{2} \)
43 \( 1 + 0.0731T + 43T^{2} \)
47 \( 1 - 10.2iT - 47T^{2} \)
53 \( 1 - 6.80T + 53T^{2} \)
59 \( 1 - 10.8iT - 59T^{2} \)
61 \( 1 + 1.63T + 61T^{2} \)
67 \( 1 - 12.8iT - 67T^{2} \)
71 \( 1 - 7.88iT - 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 4.48iT - 83T^{2} \)
89 \( 1 + 5.02iT - 89T^{2} \)
97 \( 1 - 17.0iT - 97T^{2} \)
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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

−11.16060430488892012409136573020, −10.58958698034810221793571833664, −8.923186500093962902362193950793, −8.586599754687066483535865806897, −7.12826077532058864483212221055, −5.96201935843958001813236880013, −5.59985500373040459701351175991, −4.00267950361931100694167825101, −2.59641348079322211643219466342, −0.875967632115337701482663071860, 1.96408967457189775180686703842, 3.28540038658611131938015598175, 4.81765386538637562379884925147, 6.13924112576351535941004108477, 6.62936942281323946540703019820, 7.46595567570110425831621494537, 8.595976318931198206007776006274, 10.06997355175509500982898638621, 10.72515186661160330102905597930, 11.39652893640043969653805959589

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