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Properties

Label 2-403-13.9-c1-0-19
Degree $2$
Conductor $403$
Sign $0.687 - 0.726i$
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  + (−0.399 + 0.692i)2-s + (0.607 − 1.05i)3-s + (0.680 + 1.17i)4-s + 2.33·5-s + (0.486 + 0.842i)6-s + (−0.0878 − 0.152i)7-s − 2.68·8-s + (0.760 + 1.31i)9-s + (−0.932 + 1.61i)10-s + (−0.117 + 0.203i)11-s + 1.65·12-s + (1.90 + 3.05i)13-s + 0.140·14-s + (1.41 − 2.45i)15-s + (−0.285 + 0.494i)16-s + (−0.441 − 0.764i)17-s + ⋯
L(s)  = 1  + (−0.282 + 0.489i)2-s + (0.350 − 0.607i)3-s + (0.340 + 0.589i)4-s + 1.04·5-s + (0.198 + 0.343i)6-s + (−0.0332 − 0.0575i)7-s − 0.950·8-s + (0.253 + 0.439i)9-s + (−0.295 + 0.511i)10-s + (−0.0355 + 0.0615i)11-s + 0.477·12-s + (0.528 + 0.848i)13-s + 0.0375·14-s + (0.366 − 0.634i)15-s + (−0.0714 + 0.123i)16-s + (−0.107 − 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.687 - 0.726i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.687 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54100 + 0.663095i\)
\(L(\frac12)\) \(\approx\) \(1.54100 + 0.663095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.90 - 3.05i)T \)
31 \( 1 + T \)
good2 \( 1 + (0.399 - 0.692i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.607 + 1.05i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.33T + 5T^{2} \)
7 \( 1 + (0.0878 + 0.152i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.117 - 0.203i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.441 + 0.764i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.99 + 3.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.45 + 7.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.01 - 8.69i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (0.762 - 1.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.72 + 4.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.59 - 9.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 8.73T + 53T^{2} \)
59 \( 1 + (2.31 + 4.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.99 + 6.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.82 - 3.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.53 + 11.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 - 4.47T + 83T^{2} \)
89 \( 1 + (-3.71 + 6.42i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.33 + 9.24i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.28371202911008162718003366256, −10.51178087941749246501491553316, −9.079476587240770005942158258195, −8.766969571979600476579995363583, −7.47501632185399108749429611726, −6.85184555162954771538475086611, −6.02922287792408179246488346198, −4.58621638341228049786105303671, −2.87783306204076572183221856425, −1.85437397594209026133423422358, 1.38563783404752122678613056387, 2.71910801314095431224178339705, 3.94202754825077737895289246539, 5.66915040313459411944381768072, 5.99779863129874866408666518376, 7.45723960428742370489928943778, 8.931886747182915557173995865800, 9.457826383502682793349282017616, 10.21197646703574629276022481788, 10.79639837170239695660163355355

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