Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.255 + 0.966i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.49·2-s + (−1.44 + 2.49i)3-s + 4.21·4-s + (0.778 + 1.34i)5-s + (3.59 − 6.22i)6-s + (−0.277 + 0.481i)7-s − 5.52·8-s + (−2.65 − 4.59i)9-s + (−1.94 − 3.36i)10-s + (−1.79 − 3.11i)11-s + (−6.07 + 10.5i)12-s + (−0.5 − 0.866i)13-s + (0.692 − 1.19i)14-s − 4.48·15-s + 5.34·16-s + (−1.01 + 1.75i)17-s + ⋯
L(s)  = 1  − 1.76·2-s + (−0.831 + 1.44i)3-s + 2.10·4-s + (0.348 + 0.603i)5-s + (1.46 − 2.53i)6-s + (−0.105 + 0.181i)7-s − 1.95·8-s + (−0.883 − 1.53i)9-s + (−0.613 − 1.06i)10-s + (−0.541 − 0.937i)11-s + (−1.75 + 3.03i)12-s + (−0.138 − 0.240i)13-s + (0.185 − 0.320i)14-s − 1.15·15-s + 1.33·16-s + (−0.245 + 0.425i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.255 + 0.966i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.255 + 0.966i)\)
\(L(1)\)  \(\approx\)  \(0.0583378 - 0.0449459i\)
\(L(\frac12)\)  \(\approx\)  \(0.0583378 - 0.0449459i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2.10 + 5.15i)T \)
good2 \( 1 + 2.49T + 2T^{2} \)
3 \( 1 + (1.44 - 2.49i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.778 - 1.34i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.277 - 0.481i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.79 + 3.11i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.01 - 1.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.578 - 1.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.57T + 23T^{2} \)
29 \( 1 + 4.00T + 29T^{2} \)
37 \( 1 + (-4.06 + 7.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.94 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.78 - 3.09i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 + (-4.10 - 7.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.11 - 3.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 0.963T + 61T^{2} \)
67 \( 1 + (2.74 + 4.74i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.95 + 3.38i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.90 - 5.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.38 - 5.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.967 + 1.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.41T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−10.63264514195022299768653885538, −10.26244747762265033154618724842, −9.482994907772485091066913583867, −8.624852135678100389698426552448, −7.63270420459387619629634104454, −6.21059359346036814910573820090, −5.71591003947702412653631988457, −3.96813140823285851497359012062, −2.47276794857448868581971536384, −0.094497343625998001314373657846, 1.37034640457333119587453702330, 2.25842550537183813263395159371, 5.04305281985333095388076560472, 6.30207564849889292241543856242, 7.01300846244492649715135451137, 7.76134227638552169144350477924, 8.541844011405943239484311190628, 9.686694349020406356052718806564, 10.33956695158355293298296945862, 11.50092674482898475642807879579

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