Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.596 - 0.802i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.708·2-s + (−0.837 + 1.45i)3-s − 1.49·4-s + (1.10 + 1.90i)5-s + (0.593 − 1.02i)6-s + (1.50 − 2.59i)7-s + 2.47·8-s + (0.0955 + 0.165i)9-s + (−0.779 − 1.35i)10-s + (2.27 + 3.94i)11-s + (1.25 − 2.17i)12-s + (−0.5 − 0.866i)13-s + (−1.06 + 1.84i)14-s − 3.68·15-s + 1.23·16-s + (−3.32 + 5.76i)17-s + ⋯
L(s)  = 1  − 0.501·2-s + (−0.483 + 0.837i)3-s − 0.748·4-s + (0.492 + 0.852i)5-s + (0.242 − 0.419i)6-s + (0.567 − 0.982i)7-s + 0.876·8-s + (0.0318 + 0.0551i)9-s + (−0.246 − 0.427i)10-s + (0.686 + 1.18i)11-s + (0.362 − 0.627i)12-s + (−0.138 − 0.240i)13-s + (−0.284 + 0.492i)14-s − 0.952·15-s + 0.309·16-s + (−0.806 + 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.596 - 0.802i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.596 - 0.802i)\)
\(L(1)\)  \(\approx\)  \(0.341347 + 0.679419i\)
\(L(\frac12)\)  \(\approx\)  \(0.341347 + 0.679419i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.0357 - 5.56i)T \)
good2 \( 1 + 0.708T + 2T^{2} \)
3 \( 1 + (0.837 - 1.45i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.10 - 1.90i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.50 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.27 - 3.94i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.32 - 5.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.733 + 1.26i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.58T + 23T^{2} \)
29 \( 1 + 5.25T + 29T^{2} \)
37 \( 1 + (-1.42 + 2.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.27 - 5.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.10 + 1.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + (-5.14 - 8.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.65 + 4.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + (0.702 + 1.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.94 - 8.56i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.60 - 2.77i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.38 + 2.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.35 + 12.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.89T + 89T^{2} \)
97 \( 1 + 0.237T + 97T^{2} \)
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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

−11.00540752821560737056255093169, −10.50088031773377643560526893914, −10.00488707425165046572418044415, −9.135030659940491978303624819923, −7.87202437650582961788493274621, −7.05345956898963220463423099916, −5.76876663546926684216577026139, −4.41320068163002811989756056711, −4.09125545732207831109072236749, −1.79617243844052592378229575332, 0.67114498117019022897724990599, 1.94973495824527024682038609963, 4.11866004065963831669485722660, 5.38158413743439815088905261635, 5.97340737602969305962999889278, 7.34692242051489354518610849461, 8.391790463454130156570814065960, 9.116108850619539035508304864923, 9.595309483754429846312628734496, 11.18322966975924346392548274246

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