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Properties

Label 2-403-13.3-c1-0-14
Degree $2$
Conductor $403$
Sign $0.890 - 0.454i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 1.40i)2-s + (−0.263 − 0.456i)3-s + (−0.311 + 0.539i)4-s − 1.44·5-s + (0.426 − 0.739i)6-s + (1.00 − 1.73i)7-s + 2.22·8-s + (1.36 − 2.35i)9-s + (−1.16 − 2.02i)10-s + (0.723 + 1.25i)11-s + 0.328·12-s + (2.39 + 2.69i)13-s + 3.24·14-s + (0.380 + 0.659i)15-s + (2.42 + 4.20i)16-s + (3.39 − 5.87i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.991i)2-s + (−0.152 − 0.263i)3-s + (−0.155 + 0.269i)4-s − 0.645·5-s + (0.174 − 0.301i)6-s + (0.378 − 0.655i)7-s + 0.788·8-s + (0.453 − 0.785i)9-s + (−0.369 − 0.640i)10-s + (0.218 + 0.377i)11-s + 0.0948·12-s + (0.665 + 0.746i)13-s + 0.866·14-s + (0.0982 + 0.170i)15-s + (0.607 + 1.05i)16-s + (0.822 − 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82607 + 0.439289i\)
\(L(\frac12)\) \(\approx\) \(1.82607 + 0.439289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.39 - 2.69i)T \)
31 \( 1 - T \)
good2 \( 1 + (-0.809 - 1.40i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.263 + 0.456i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
7 \( 1 + (-1.00 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.723 - 1.25i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.39 + 5.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.87 - 3.25i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.430 - 0.745i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.627 - 1.08i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-2.56 - 4.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.36 + 7.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.84 + 8.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.35T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + (0.770 - 1.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.62 - 4.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.04 - 12.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.77 - 6.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 5.48T + 79T^{2} \)
83 \( 1 - 2.29T + 83T^{2} \)
89 \( 1 + (-0.376 - 0.651i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.03 - 3.51i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.56367034981722926388802377182, −10.46556670475481030452440055603, −9.469950297033866198614465070040, −8.142485106219348461731635153211, −7.28418263848629104497572110814, −6.77398626152755876999466028836, −5.69068749924217196631076871998, −4.47485972842254140448395892153, −3.75606430234616240937422442712, −1.35118295754746589189090305356, 1.67028448592831107863458969535, 3.09500503818828864840380367447, 4.08853230635609842619061512597, 5.01005139251653326509488399684, 6.18281398668168517398971542058, 7.86639107150487766554658901307, 8.186802184832746787635388443561, 9.708699429936644640220375197046, 10.79636079337849663124206330729, 11.09708639195286224739805011749

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