Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.591 - 0.806i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.21i·2-s − 2.59·3-s − 2.91·4-s − 1.00i·5-s − 5.74i·6-s − 1.82i·7-s − 2.02i·8-s + 3.71·9-s + 2.22·10-s − 2.73i·11-s + 7.55·12-s + (2.90 + 2.13i)13-s + 4.03·14-s + 2.60i·15-s − 1.33·16-s + 2.66·17-s + ⋯
L(s)  = 1  + 1.56i·2-s − 1.49·3-s − 1.45·4-s − 0.449i·5-s − 2.34i·6-s − 0.688i·7-s − 0.716i·8-s + 1.23·9-s + 0.705·10-s − 0.824i·11-s + 2.18·12-s + (0.806 + 0.591i)13-s + 1.07·14-s + 0.673i·15-s − 0.334·16-s + 0.647·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.591 - 0.806i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.591 - 0.806i)$
$L(1)$  $\approx$  $0.670120 + 0.339623i$
$L(\frac12)$  $\approx$  $0.670120 + 0.339623i$
$L(\frac{3}{2})$   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 \( 1 + (-2.90 - 2.13i)T \)
31 \( 1 - iT \)
good2 \( 1 - 2.21iT - 2T^{2} \)
3 \( 1 + 2.59T + 3T^{2} \)
5 \( 1 + 1.00iT - 5T^{2} \)
7 \( 1 + 1.82iT - 7T^{2} \)
11 \( 1 + 2.73iT - 11T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 + 2.04iT - 19T^{2} \)
23 \( 1 - 1.36T + 23T^{2} \)
29 \( 1 + 3.75T + 29T^{2} \)
37 \( 1 + 5.04iT - 37T^{2} \)
41 \( 1 - 3.02iT - 41T^{2} \)
43 \( 1 - 9.12T + 43T^{2} \)
47 \( 1 + 7.62iT - 47T^{2} \)
53 \( 1 - 5.26T + 53T^{2} \)
59 \( 1 + 2.66iT - 59T^{2} \)
61 \( 1 - 7.14T + 61T^{2} \)
67 \( 1 + 9.48iT - 67T^{2} \)
71 \( 1 + 4.99iT - 71T^{2} \)
73 \( 1 + 4.50iT - 73T^{2} \)
79 \( 1 + 0.813T + 79T^{2} \)
83 \( 1 - 1.18iT - 83T^{2} \)
89 \( 1 - 16.2iT - 89T^{2} \)
97 \( 1 - 3.69iT - 97T^{2} \)
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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

−11.22893866816759510091970665949, −10.72763518735831511912982314270, −9.310814124760996536490735449574, −8.436826751705589305169953329920, −7.31123291483485580078860569450, −6.57760959484391883205689000271, −5.75596208408127826239576749685, −5.08827481971264526280415277005, −4.00955705247480575518946125957, −0.75473608244935910638462840413, 1.16058977129461835611252743019, 2.69188731188820636254561960905, 4.01477904066591247004045890525, 5.21161746582651203037915500906, 6.06976899128578159639708529528, 7.24144093708370999440165267040, 8.771479240541126962395370318922, 9.894134163109172281528840818511, 10.50954244003541959547019269820, 11.18334458100207707168952020078

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