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  :  $\chi_{403} (118, \cdot )$
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

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

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