Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.628 - 0.777i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 0.291·2-s + (−0.965 − 1.67i)3-s − 1.91·4-s + (−0.817 + 1.41i)5-s + (0.281 + 0.487i)6-s + (−0.566 − 0.980i)7-s + 1.14·8-s + (−0.363 + 0.629i)9-s + (0.238 − 0.412i)10-s + (−1.40 + 2.43i)11-s + (1.84 + 3.20i)12-s + (−0.5 + 0.866i)13-s + (0.164 + 0.285i)14-s + 3.15·15-s + 3.49·16-s + (−0.697 − 1.20i)17-s + ⋯
L(s)  = 1  − 0.206·2-s + (−0.557 − 0.965i)3-s − 0.957·4-s + (−0.365 + 0.633i)5-s + (0.114 + 0.198i)6-s + (−0.213 − 0.370i)7-s + 0.403·8-s + (−0.121 + 0.209i)9-s + (0.0753 − 0.130i)10-s + (−0.424 + 0.734i)11-s + (0.533 + 0.924i)12-s + (−0.138 + 0.240i)13-s + (0.0440 + 0.0763i)14-s + 0.815·15-s + 0.874·16-s + (−0.169 − 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.628 - 0.777i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.628 - 0.777i)\)
\(L(1)\)  \(\approx\)  \(0.486615 + 0.232372i\)
\(L(\frac12)\)  \(\approx\)  \(0.486615 + 0.232372i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.187 - 5.56i)T \)
good2 \( 1 + 0.291T + 2T^{2} \)
3 \( 1 + (0.965 + 1.67i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.817 - 1.41i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.566 + 0.980i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.40 - 2.43i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.697 + 1.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.07 - 7.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.36T + 23T^{2} \)
29 \( 1 - 8.21T + 29T^{2} \)
37 \( 1 + (-0.953 - 1.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.861 + 1.49i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.941 + 1.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.80T + 47T^{2} \)
53 \( 1 + (6.20 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.48 + 2.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.73T + 61T^{2} \)
67 \( 1 + (3.58 - 6.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.99 - 8.64i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.66 - 13.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.12 + 7.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.77 - 9.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.04T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.54362584603499873497769987943, −10.32325115351342745980907624928, −9.784440273859592462085765057247, −8.485439324459713236956277816517, −7.46527544390516604843069643633, −6.96267541022639175312982518371, −5.73217993029784631894031942306, −4.56366221344216386849535116103, −3.28440293865994957252615314237, −1.29146094204099107563086214372, 0.47981980884635773509214178351, 3.17284250618345385510250013069, 4.60466108713366501502915497797, 4.92649189771603319394575493458, 6.04708280481582011420356388120, 7.70951305746885037809487745747, 8.621753307048475418765854068654, 9.329467198767528684123727123122, 10.14010298631160921312263356155, 11.00665288848640113124020138703

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