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Properties

Label 2-403-13.10-c1-0-20
Degree $2$
Conductor $403$
Sign $0.921 + 0.388i$
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  + (1.84 − 1.06i)2-s + (−0.224 − 0.389i)3-s + (1.26 − 2.19i)4-s + 3.56i·5-s + (−0.828 − 0.478i)6-s + (2.73 + 1.57i)7-s − 1.12i·8-s + (1.39 − 2.42i)9-s + (3.79 + 6.57i)10-s + (−1.51 + 0.875i)11-s − 1.13·12-s + (−0.491 − 3.57i)13-s + 6.72·14-s + (1.38 − 0.802i)15-s + (1.33 + 2.30i)16-s + (0.344 − 0.596i)17-s + ⋯
L(s)  = 1  + (1.30 − 0.752i)2-s + (−0.129 − 0.224i)3-s + (0.632 − 1.09i)4-s + 1.59i·5-s + (−0.338 − 0.195i)6-s + (1.03 + 0.597i)7-s − 0.397i·8-s + (0.466 − 0.807i)9-s + (1.20 + 2.07i)10-s + (−0.457 + 0.263i)11-s − 0.328·12-s + (−0.136 − 0.990i)13-s + 1.79·14-s + (0.358 − 0.207i)15-s + (0.332 + 0.576i)16-s + (0.0835 − 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.921 + 0.388i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.921 + 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69145 - 0.544945i\)
\(L(\frac12)\) \(\approx\) \(2.69145 - 0.544945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.491 + 3.57i)T \)
31 \( 1 + iT \)
good2 \( 1 + (-1.84 + 1.06i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.224 + 0.389i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.56iT - 5T^{2} \)
7 \( 1 + (-2.73 - 1.57i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.51 - 0.875i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.344 + 0.596i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.77 + 2.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.160 - 0.277i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.04 + 3.53i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-3.42 + 1.97i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.85 - 4.53i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.24 + 9.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.36iT - 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + (-8.72 - 5.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.42 - 4.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.93 - 3.42i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.76 - 3.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.83iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 5.52iT - 83T^{2} \)
89 \( 1 + (4.70 - 2.71i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.85 + 1.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.31239585661640164590535654587, −10.74400187931977310018551041021, −9.878223817774174861040857869775, −8.271542085714242875632915361287, −7.22096034286836153417338415488, −6.19050519638771335567168064578, −5.30034246817862935274196697187, −4.08942684677393121164051072747, −2.96552883331037082864068362333, −2.10568735749321688775687233530, 1.64148435334681269457584422775, 3.96239465613143757031404006044, 4.79529104486679968583337101194, 5.02304867103035711444043836712, 6.29521868352020563321694641396, 7.62110139060298370571044209190, 8.188696008975453640914689975490, 9.375666687195861720207092083924, 10.60115640312615090408363708557, 11.58066617522452158621763708967

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