Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.817 + 0.576i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.937i·2-s + 1.68·3-s + 1.12·4-s + 0.290i·5-s − 1.58i·6-s + 3.85i·7-s − 2.92i·8-s − 0.158·9-s + 0.272·10-s − 2.09i·11-s + 1.89·12-s + (2.07 − 2.94i)13-s + 3.61·14-s + 0.490i·15-s − 0.499·16-s + 5.29·17-s + ⋯
L(s)  = 1  − 0.662i·2-s + 0.973·3-s + 0.560·4-s + 0.130i·5-s − 0.645i·6-s + 1.45i·7-s − 1.03i·8-s − 0.0527·9-s + 0.0861·10-s − 0.632i·11-s + 0.545·12-s + (0.576 − 0.817i)13-s + 0.965·14-s + 0.126i·15-s − 0.124·16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.817 + 0.576i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.817 + 0.576i)$
$L(1)$  $\approx$  $2.04100 - 0.646957i$
$L(\frac12)$  $\approx$  $2.04100 - 0.646957i$
$L(\frac{3}{2})$   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 \( 1 + (-2.07 + 2.94i)T \)
31 \( 1 + iT \)
good2 \( 1 + 0.937iT - 2T^{2} \)
3 \( 1 - 1.68T + 3T^{2} \)
5 \( 1 - 0.290iT - 5T^{2} \)
7 \( 1 - 3.85iT - 7T^{2} \)
11 \( 1 + 2.09iT - 11T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 3.22iT - 19T^{2} \)
23 \( 1 + 6.80T + 23T^{2} \)
29 \( 1 + 4.52T + 29T^{2} \)
37 \( 1 - 4.81iT - 37T^{2} \)
41 \( 1 - 6.15iT - 41T^{2} \)
43 \( 1 + 7.12T + 43T^{2} \)
47 \( 1 + 4.76iT - 47T^{2} \)
53 \( 1 + 5.08T + 53T^{2} \)
59 \( 1 + 9.83iT - 59T^{2} \)
61 \( 1 + 9.60T + 61T^{2} \)
67 \( 1 - 6.38iT - 67T^{2} \)
71 \( 1 + 3.60iT - 71T^{2} \)
73 \( 1 + 8.21iT - 73T^{2} \)
79 \( 1 + 4.16T + 79T^{2} \)
83 \( 1 - 5.60iT - 83T^{2} \)
89 \( 1 - 5.99iT - 89T^{2} \)
97 \( 1 - 1.49iT - 97T^{2} \)
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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

−11.29599988111624199773433409631, −10.22685355819748112103650507722, −9.441192303633398885409349274217, −8.348725136765506881213527825386, −7.890793228335441264951288157519, −6.24551023713650080473544577260, −5.56823792637907805887719663850, −3.45260527135599173631401983850, −2.98929164506142434449691566803, −1.79338321221897472760009807780, 1.76924368451921999607736763013, 3.24898328087774399693823349449, 4.34867810784165120888469837190, 5.78558599460495880354788630895, 6.98265380884534706744502142697, 7.53353429714858737865454725772, 8.353169250277200322350766192728, 9.379316687138301521349422541572, 10.40213482320496857543859651341, 11.24072580033414424177038836572

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