Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.979 + 0.202i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.272·2-s + (0.850 − 1.47i)3-s − 1.92·4-s + (1.58 + 2.74i)5-s + (−0.231 + 0.400i)6-s + (1.18 − 2.04i)7-s + 1.06·8-s + (0.0537 + 0.0931i)9-s + (−0.431 − 0.747i)10-s + (1.18 + 2.04i)11-s + (−1.63 + 2.83i)12-s + (−0.5 − 0.866i)13-s + (−0.321 + 0.557i)14-s + 5.39·15-s + 3.56·16-s + (2.76 − 4.79i)17-s + ⋯
L(s)  = 1  − 0.192·2-s + (0.490 − 0.850i)3-s − 0.962·4-s + (0.709 + 1.22i)5-s + (−0.0944 + 0.163i)6-s + (0.446 − 0.773i)7-s + 0.377·8-s + (0.0179 + 0.0310i)9-s + (−0.136 − 0.236i)10-s + (0.355 + 0.616i)11-s + (−0.472 + 0.818i)12-s + (−0.138 − 0.240i)13-s + (−0.0859 + 0.148i)14-s + 1.39·15-s + 0.890·16-s + (0.671 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.979 + 0.202i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.979 + 0.202i)\)
\(L(1)\)  \(\approx\)  \(1.45288 - 0.148833i\)
\(L(\frac12)\)  \(\approx\)  \(1.45288 - 0.148833i\)
\(L(\frac{3}{2})\)   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 \( 1 + (3.67 + 4.18i)T \)
good2 \( 1 + 0.272T + 2T^{2} \)
3 \( 1 + (-0.850 + 1.47i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.58 - 2.74i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.18 + 2.04i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.18 - 2.04i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.76 + 4.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.29 - 5.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.73T + 23T^{2} \)
29 \( 1 - 3.62T + 29T^{2} \)
37 \( 1 + (-2.79 + 4.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.02 + 5.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.343 + 0.595i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.64T + 47T^{2} \)
53 \( 1 + (1.25 + 2.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.82 - 10.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 5.03T + 61T^{2} \)
67 \( 1 + (-7.51 - 13.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.77 - 11.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.59 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.81 - 15.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.394 - 0.683i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 2.61T + 97T^{2} \)
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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

−10.97076175179620773370787093690, −10.21325866202900936692943093121, −9.545850455764863317009193946914, −8.319904796602938515806103681923, −7.42844029069305408838764885036, −6.90929846587462294342604826338, −5.49694038860243040911396307560, −4.21832478355784083692767985376, −2.82077059166414296602472358618, −1.43658693629296870248070333865, 1.33954784498907621115743387041, 3.30735829188075685163281539678, 4.68533650247049306712913614330, 5.03342274791269249876259493187, 6.32388850370109052485542675219, 8.288065168590176184246219832552, 8.768083946754900139899698938786, 9.243443287928947397006672339195, 10.00225271334919717199393977462, 11.11028321062851779520026117653

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