Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.0444 + 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.53·2-s + (−1.60 − 2.78i)3-s + 4.43·4-s + (1.34 − 2.33i)5-s + (−4.07 − 7.05i)6-s + (0.661 + 1.14i)7-s + 6.17·8-s + (−3.65 + 6.33i)9-s + (3.42 − 5.92i)10-s + (−1.96 + 3.39i)11-s + (−7.11 − 12.3i)12-s + (−0.5 + 0.866i)13-s + (1.67 + 2.90i)14-s − 8.66·15-s + 6.78·16-s + (−1.50 − 2.61i)17-s + ⋯
L(s)  = 1  + 1.79·2-s + (−0.926 − 1.60i)3-s + 2.21·4-s + (0.603 − 1.04i)5-s + (−1.66 − 2.87i)6-s + (0.250 + 0.433i)7-s + 2.18·8-s + (−1.21 + 2.11i)9-s + (1.08 − 1.87i)10-s + (−0.591 + 1.02i)11-s + (−2.05 − 3.55i)12-s + (−0.138 + 0.240i)13-s + (0.448 + 0.776i)14-s − 2.23·15-s + 1.69·16-s + (−0.366 − 0.633i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0444 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0444 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.0444 + 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.0444 + 0.999i)\)
\(L(1)\)  \(\approx\)  \(2.19203 - 2.09668i\)
\(L(\frac12)\)  \(\approx\)  \(2.19203 - 2.09668i\)
\(L(\frac{3}{2})\)   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 \( 1 + (3.54 + 4.29i)T \)
good2 \( 1 - 2.53T + 2T^{2} \)
3 \( 1 + (1.60 + 2.78i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.34 + 2.33i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.661 - 1.14i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.96 - 3.39i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.50 + 2.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.04 - 3.53i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
37 \( 1 + (5.25 + 9.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.201 - 0.348i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.37 - 9.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.91T + 47T^{2} \)
53 \( 1 + (1.53 - 2.65i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.36 - 2.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 7.35T + 61T^{2} \)
67 \( 1 + (6.80 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.12 + 7.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.34 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.53 + 6.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.35 + 2.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 1.84T + 97T^{2} \)
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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

−11.64334494069061187116772690546, −10.79162851591749465736905143587, −9.135408542199781423954038777263, −7.56937196644178032210344500554, −7.05760411240165561952225864164, −5.80236532112255965446609155363, −5.41402072657713823008426501486, −4.61971166469020058212601384789, −2.47925878829557831741006595651, −1.58885843782493132716816458317, 2.92055005954697061849194023958, 3.58695892998209804489726973164, 4.79225993858903071943939080063, 5.42824013054112010059410466369, 6.22021663899664737127034328672, 7.09135581248852021544139141598, 9.030497587344547079899517304636, 10.41603732749048747358972572226, 10.81967961485111298426579878123, 11.20459016321042831863392383271

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