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Properties

Label 2-403-13.4-c1-0-14
Degree $2$
Conductor $403$
Sign $0.202 + 0.979i$
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  + (−1.56 − 0.901i)2-s + (0.553 − 0.959i)3-s + (0.626 + 1.08i)4-s − 0.771i·5-s + (−1.73 + 0.999i)6-s + (3.80 − 2.19i)7-s + 1.34i·8-s + (0.886 + 1.53i)9-s + (−0.695 + 1.20i)10-s + (4.70 + 2.71i)11-s + 1.38·12-s + (3.40 + 1.17i)13-s − 7.92·14-s + (−0.740 − 0.427i)15-s + (2.46 − 4.27i)16-s + (−0.585 − 1.01i)17-s + ⋯
L(s)  = 1  + (−1.10 − 0.637i)2-s + (0.319 − 0.553i)3-s + (0.313 + 0.542i)4-s − 0.345i·5-s + (−0.706 + 0.407i)6-s + (1.43 − 0.830i)7-s + 0.476i·8-s + (0.295 + 0.511i)9-s + (−0.220 + 0.381i)10-s + (1.41 + 0.818i)11-s + 0.400·12-s + (0.945 + 0.326i)13-s − 2.11·14-s + (−0.191 − 0.110i)15-s + (0.616 − 1.06i)16-s + (−0.142 − 0.246i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.202 + 0.979i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (342, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.202 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846306 - 0.689521i\)
\(L(\frac12)\) \(\approx\) \(0.846306 - 0.689521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.40 - 1.17i)T \)
31 \( 1 + iT \)
good2 \( 1 + (1.56 + 0.901i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.553 + 0.959i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.771iT - 5T^{2} \)
7 \( 1 + (-3.80 + 2.19i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.70 - 2.71i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.585 + 1.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.24 - 3.02i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.575 - 0.996i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.41 - 4.17i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (5.52 + 3.18i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.1 + 5.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.82 + 4.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.06iT - 47T^{2} \)
53 \( 1 - 1.26T + 53T^{2} \)
59 \( 1 + (7.47 - 4.31i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.487 + 0.844i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.34 + 4.24i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.32 - 1.92i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.37iT - 73T^{2} \)
79 \( 1 - 8.38T + 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + (9.37 + 5.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.0599 + 0.0345i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.83335940413022836167800901691, −10.35445057946097068275578352048, −8.989515642837606839305808543830, −8.550213622847257264885165908103, −7.63003885502945817314415931541, −6.79595017372616888452468182222, −5.02007024417229311632672784879, −4.01490101104624532344633921833, −1.75690470855095026118370125860, −1.48012560170630946735585892190, 1.41502640410719941356423933035, 3.43550276147532241350116515348, 4.54239897891104658720641444268, 6.14440763428301515804339924916, 6.78537168287141690189860601948, 8.310491608540093866864451429720, 8.593610660514430542905553756145, 9.235499687542590923559164700878, 10.40528636559303750207166587341, 11.19330867182876319084589603124

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