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  :  $\chi_{403} (118, \cdot )$
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

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

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