Properties

Label 2-403-403.4-c1-0-15
Degree $2$
Conductor $403$
Sign $-0.271 - 0.962i$
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  + (−0.117 + 0.551i)2-s + (1.94 + 2.15i)3-s + (1.53 + 0.684i)4-s − 0.526i·5-s + (−1.41 + 0.817i)6-s + (0.000659 − 0.00148i)7-s + (−1.21 + 1.67i)8-s + (−0.567 + 5.39i)9-s + (0.289 + 0.0616i)10-s + (−4.57 + 0.480i)11-s + (1.50 + 4.64i)12-s + (−2.33 − 2.74i)13-s + (0.000738 + 0.000536i)14-s + (1.13 − 1.02i)15-s + (1.46 + 1.63i)16-s + (0.708 − 6.74i)17-s + ⋯
L(s)  = 1  + (−0.0828 + 0.389i)2-s + (1.12 + 1.24i)3-s + (0.768 + 0.342i)4-s − 0.235i·5-s + (−0.578 + 0.333i)6-s + (0.000249 − 0.000559i)7-s + (−0.431 + 0.593i)8-s + (−0.189 + 1.79i)9-s + (0.0916 + 0.0194i)10-s + (−1.37 + 0.145i)11-s + (0.435 + 1.34i)12-s + (−0.647 − 0.761i)13-s + (0.000197 + 0.000143i)14-s + (0.293 − 0.263i)15-s + (0.367 + 0.408i)16-s + (0.171 − 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.271 - 0.962i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.271 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24290 + 1.64246i\)
\(L(\frac12)\) \(\approx\) \(1.24290 + 1.64246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.33 + 2.74i)T \)
31 \( 1 + (1.79 + 5.26i)T \)
good2 \( 1 + (0.117 - 0.551i)T + (-1.82 - 0.813i)T^{2} \)
3 \( 1 + (-1.94 - 2.15i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + 0.526iT - 5T^{2} \)
7 \( 1 + (-0.000659 + 0.00148i)T + (-4.68 - 5.20i)T^{2} \)
11 \( 1 + (4.57 - 0.480i)T + (10.7 - 2.28i)T^{2} \)
17 \( 1 + (-0.708 + 6.74i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-4.55 - 4.10i)T + (1.98 + 18.8i)T^{2} \)
23 \( 1 + (-5.24 + 2.33i)T + (15.3 - 17.0i)T^{2} \)
29 \( 1 + (-0.908 - 0.193i)T + (26.4 + 11.7i)T^{2} \)
37 \( 1 + (-1.81 - 1.04i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.48 + 6.99i)T + (-37.4 - 16.6i)T^{2} \)
43 \( 1 + (7.41 - 8.23i)T + (-4.49 - 42.7i)T^{2} \)
47 \( 1 + (4.40 + 1.43i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.804 - 0.584i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.65 - 7.77i)T + (-53.8 + 23.9i)T^{2} \)
61 \( 1 + (5.48 + 9.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 5.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.6 + 1.11i)T + (69.4 + 14.7i)T^{2} \)
73 \( 1 + (2.48 + 3.41i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (11.7 + 8.54i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.341 + 0.110i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (2.96 - 0.311i)T + (87.0 - 18.5i)T^{2} \)
97 \( 1 + (-4.76 + 10.6i)T + (-64.9 - 72.0i)T^{2} \)
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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

−11.30957253364889476351763371013, −10.38166363702695813275516991845, −9.713672950129586313313126795653, −8.733777471224746229414940778911, −7.82657986647760292801947492619, −7.30916869232548066361751894663, −5.48129817948315005373511674858, −4.77719416792861653894714942638, −3.09045519635775995397446909003, −2.70495538725951507705962730103, 1.40795488013412299303121451949, 2.54161677525680967001085889030, 3.24216552459538810052937366929, 5.31562624937788632881208573167, 6.72300063689133062945555276718, 7.17844348698657797613924370227, 8.122336036101600986486965086075, 9.074807157707268011596153897548, 10.14646120505672594417577116500, 11.03235893074050175826335850615

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