Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.832 + 0.554i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·4-s − 4i·5-s + 2i·7-s + 9-s i·11-s + 4·12-s + (−2 + 3i)13-s − 8i·15-s + 4·16-s − 2·17-s + 2i·19-s − 8i·20-s + 4i·21-s − 4·23-s + ⋯
L(s)  = 1  + 1.15·3-s + 4-s − 1.78i·5-s + 0.755i·7-s + 0.333·9-s − 0.301i·11-s + 1.15·12-s + (−0.554 + 0.832i)13-s − 2.06i·15-s + 16-s − 0.485·17-s + 0.458i·19-s − 1.78i·20-s + 0.872i·21-s − 0.834·23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.832 + 0.554i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.832 + 0.554i)$
$L(1)$  $\approx$  $2.17962 - 0.659937i$
$L(\frac12)$  $\approx$  $2.17962 - 0.659937i$
$L(\frac{3}{2})$   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 \( 1 + (2 - 3i)T \)
31 \( 1 - iT \)
good2 \( 1 - 2T^{2} \)
3 \( 1 - 2T + 3T^{2} \)
5 \( 1 + 4iT - 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + iT - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 15iT - 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−11.48824188220044485522628631094, −9.950788799957586492306938655011, −9.148093643779623950381364834693, −8.424697193241376930589622367273, −7.87332931386543365109413609431, −6.42619710812602113355812489125, −5.34892375760208459795988448951, −4.15389208039854490017879517281, −2.67639293080402667122026786589, −1.67317628191153079059259795335, 2.33898214138781229335548978487, 2.87913545534096285272776893375, 3.91813201613630258886152375650, 5.93747167203652307881585637367, 7.03246694791877821157741815824, 7.39411504380701027906515046948, 8.330891792706934249857488501637, 9.865101921932921833798347143972, 10.40650301407362823743526944467, 11.11640169391882038432642624454

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