Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.135 - 0.990i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.475i·2-s + 0.345·3-s + 1.77·4-s + 0.161i·5-s + 0.164i·6-s + 4.07i·7-s + 1.79i·8-s − 2.88·9-s − 0.0768·10-s + 3.77i·11-s + 0.613·12-s + (−3.57 − 0.487i)13-s − 1.94·14-s + 0.0558i·15-s + 2.69·16-s + 1.83·17-s + ⋯
L(s)  = 1  + 0.336i·2-s + 0.199·3-s + 0.886·4-s + 0.0722i·5-s + 0.0671i·6-s + 1.54i·7-s + 0.634i·8-s − 0.960·9-s − 0.0243·10-s + 1.13i·11-s + 0.176·12-s + (−0.990 − 0.135i)13-s − 0.518·14-s + 0.0144i·15-s + 0.673·16-s + 0.444·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.135 - 0.990i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.135 - 0.990i)$
$L(1)$  $\approx$  $1.22827 + 1.07203i$
$L(\frac12)$  $\approx$  $1.22827 + 1.07203i$
$L(\frac{3}{2})$   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 \( 1 + (3.57 + 0.487i)T \)
31 \( 1 + iT \)
good2 \( 1 - 0.475iT - 2T^{2} \)
3 \( 1 - 0.345T + 3T^{2} \)
5 \( 1 - 0.161iT - 5T^{2} \)
7 \( 1 - 4.07iT - 7T^{2} \)
11 \( 1 - 3.77iT - 11T^{2} \)
17 \( 1 - 1.83T + 17T^{2} \)
19 \( 1 + 6.07iT - 19T^{2} \)
23 \( 1 - 8.58T + 23T^{2} \)
29 \( 1 + 5.21T + 29T^{2} \)
37 \( 1 + 11.7iT - 37T^{2} \)
41 \( 1 - 4.76iT - 41T^{2} \)
43 \( 1 - 6.24T + 43T^{2} \)
47 \( 1 + 4.70iT - 47T^{2} \)
53 \( 1 - 6.16T + 53T^{2} \)
59 \( 1 - 7.98iT - 59T^{2} \)
61 \( 1 + 4.52T + 61T^{2} \)
67 \( 1 + 7.69iT - 67T^{2} \)
71 \( 1 - 7.77iT - 71T^{2} \)
73 \( 1 - 4.96iT - 73T^{2} \)
79 \( 1 - 8.26T + 79T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + 3.17iT - 89T^{2} \)
97 \( 1 + 0.366iT - 97T^{2} \)
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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

−11.51084073385104166140779900560, −10.75996221228885576310295057559, −9.334993133313735702456274576006, −8.858493622347315968041743396187, −7.57242326495139937424523199187, −6.90209009395338836922396850480, −5.65281949626883227011589714281, −5.03736984295245179091591982113, −2.84217085784167253917115284871, −2.33839181447394994229310234336, 1.08251451617496521826429578685, 2.88686809246881298936984185930, 3.67680226511296515685380892496, 5.25416188776818165346682147196, 6.44118203975022582539121551901, 7.35464784482550877880711985769, 8.129506315895779535448316035113, 9.374253691515365983075286851266, 10.48571725695364794067240817048, 10.91873591840496702579403358895

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