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Properties

Label 2-403-403.25-c1-0-11
Degree $2$
Conductor $403$
Sign $0.979 + 0.201i$
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  − 0.909i·2-s + (0.193 + 0.335i)3-s + 1.17·4-s + (3.26 + 1.88i)5-s + (0.305 − 0.176i)6-s + (−3.57 + 2.06i)7-s − 2.88i·8-s + (1.42 − 2.46i)9-s + (1.71 − 2.96i)10-s + (−0.310 − 0.179i)11-s + (0.227 + 0.394i)12-s + (2.94 + 2.07i)13-s + (1.87 + 3.25i)14-s + 1.46i·15-s − 0.279·16-s + (−0.359 − 0.622i)17-s + ⋯
L(s)  = 1  − 0.643i·2-s + (0.111 + 0.193i)3-s + 0.586·4-s + (1.45 + 0.842i)5-s + (0.124 − 0.0720i)6-s + (−1.35 + 0.780i)7-s − 1.02i·8-s + (0.474 − 0.822i)9-s + (0.542 − 0.938i)10-s + (−0.0935 − 0.0539i)11-s + (0.0656 + 0.113i)12-s + (0.817 + 0.575i)13-s + (0.501 + 0.869i)14-s + 0.377i·15-s − 0.0697·16-s + (−0.0871 − 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.979 + 0.201i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.979 + 0.201i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90703 - 0.194138i\)
\(L(\frac12)\) \(\approx\) \(1.90703 - 0.194138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.94 - 2.07i)T \)
31 \( 1 + (2.82 + 4.79i)T \)
good2 \( 1 + 0.909iT - 2T^{2} \)
3 \( 1 + (-0.193 - 0.335i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.26 - 1.88i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.57 - 2.06i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.310 + 0.179i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.359 + 0.622i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.27 - 3.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.38T + 23T^{2} \)
29 \( 1 + 5.79T + 29T^{2} \)
37 \( 1 + (-5.77 + 3.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.227 + 0.131i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.36 + 2.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.52iT - 47T^{2} \)
53 \( 1 + (1.14 - 1.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.44 - 1.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (7.25 + 4.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.34 + 4.81i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.72 - 0.993i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.43 + 5.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.1 - 6.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 - 2.44iT - 97T^{2} \)
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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

−10.97882819453092035362585479492, −10.38838990727326026950001461287, −9.385406589122323213666988998843, −9.224788651614307111010015972265, −7.11087833970679065287964660647, −6.23428299757286028973490482859, −6.02265210697474633654887262365, −3.75812070294511402186408784079, −2.86162815317702555423416509021, −1.85560544963927048683500559306, 1.55176558114427169796801739294, 2.89215379911697938671676244229, 4.71357537132926108505427806815, 5.82468083092494562837428257501, 6.49911265457738295409827680692, 7.34433949187444672932500555319, 8.528843417133578199574817165709, 9.401684736107157987236682003413, 10.42850978884034805373559248770, 10.91347733624660092020997297680

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