Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.522 - 0.852i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s − 3·5-s + (−1 + 1.73i)7-s − 3·8-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)10-s + (−2.5 + 2.59i)13-s + 1.99·14-s + (0.500 + 0.866i)16-s + (−2.5 + 4.33i)17-s − 3·18-s + (−2 + 3.46i)19-s + (−1.50 + 2.59i)20-s + (−2 − 3.46i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s − 1.34·5-s + (−0.377 + 0.654i)7-s − 1.06·8-s + (0.5 − 0.866i)9-s + (0.474 + 0.821i)10-s + (−0.693 + 0.720i)13-s + 0.534·14-s + (0.125 + 0.216i)16-s + (−0.606 + 1.05i)17-s − 0.707·18-s + (−0.458 + 0.794i)19-s + (−0.335 + 0.580i)20-s + (−0.417 − 0.722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.522 - 0.852i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (94, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.522 - 0.852i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + (2.5 - 2.59i)T \)
31 \( 1 + T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2 + 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9 + 15.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70945340154624265979136607798, −9.804730765849498770748622836713, −8.992638168339053293765512103435, −8.067710430677879380414335847624, −6.78037284061726580088351460140, −6.08134337922909349089170455968, −4.42270367648091092903441358103, −3.43795209240806247331391329874, −1.96205391458802541652082003209, 0, 2.77490665217558160198118340840, 3.94437873180279871162996668980, 5.02757091133561836856995891626, 6.67494376083675504444709883359, 7.49431526448001500136698274360, 7.75861698098793931903459896864, 8.880855987920655196567674472333, 9.996200631951439651012765452789, 11.11568724695176633663997251758

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