Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.982 - 0.186i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.55·2-s + (−1.19 + 2.07i)3-s + 0.427·4-s + (−0.908 − 1.57i)5-s + (−1.86 + 3.23i)6-s + (−1.80 + 3.12i)7-s − 2.45·8-s + (−1.37 − 2.37i)9-s + (−1.41 − 2.45i)10-s + (−0.657 − 1.13i)11-s + (−0.512 + 0.887i)12-s + (−0.5 − 0.866i)13-s + (−2.81 + 4.87i)14-s + 4.35·15-s − 4.67·16-s + (−0.822 + 1.42i)17-s + ⋯
L(s)  = 1  + 1.10·2-s + (−0.691 + 1.19i)3-s + 0.213·4-s + (−0.406 − 0.703i)5-s + (−0.762 + 1.32i)6-s + (−0.682 + 1.18i)7-s − 0.866·8-s + (−0.457 − 0.792i)9-s + (−0.447 − 0.775i)10-s + (−0.198 − 0.343i)11-s + (−0.147 + 0.256i)12-s + (−0.138 − 0.240i)13-s + (−0.751 + 1.30i)14-s + 1.12·15-s − 1.16·16-s + (−0.199 + 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.982 - 0.186i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.982 - 0.186i)\)
\(L(1)\)  \(\approx\)  \(0.0731647 + 0.778799i\)
\(L(\frac12)\)  \(\approx\)  \(0.0731647 + 0.778799i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-3.74 - 4.12i)T \)
good2 \( 1 - 1.55T + 2T^{2} \)
3 \( 1 + (1.19 - 2.07i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.908 + 1.57i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.80 - 3.12i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.657 + 1.13i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.822 - 1.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.67 - 6.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.76T + 23T^{2} \)
29 \( 1 - 0.753T + 29T^{2} \)
37 \( 1 + (-0.869 + 1.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.45 - 2.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.15 - 7.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.25T + 47T^{2} \)
53 \( 1 + (-4.95 - 8.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.40 + 4.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + (0.609 + 1.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.33 + 4.04i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.17 - 2.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.15 - 8.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.38 - 2.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−12.01256255211702214782395780996, −10.91633957790693824375248422251, −9.946958687345720344533214642818, −9.033800588914832864037962726688, −8.307211575005124986783695563708, −6.26346557688961216338980254287, −5.68798315606355615135948145840, −4.82293108800944802561752565751, −4.06245449722067142572071366908, −2.94515017444776012999834785096, 0.37558959472461373583541011086, 2.63628371631771819506238723878, 3.88202769122422766960950439387, 4.92246176786037610839231048335, 6.24902326711499001120062473143, 6.92252019748930686420638333750, 7.35786341963013462140015831996, 8.962275423818503101599272231655, 10.26355143933414437394091075873, 11.24265823488434646139186487965

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