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Properties

Label 2-403-13.4-c1-0-25
Degree $2$
Conductor $403$
Sign $0.190 + 0.981i$
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  + (0.519 + 0.299i)2-s + (0.801 − 1.38i)3-s + (−0.820 − 1.42i)4-s + 0.195i·5-s + (0.832 − 0.480i)6-s + (2.33 − 1.34i)7-s − 2.18i·8-s + (0.214 + 0.371i)9-s + (−0.0586 + 0.101i)10-s + (0.499 + 0.288i)11-s − 2.63·12-s + (−3.39 − 1.21i)13-s + 1.61·14-s + (0.271 + 0.157i)15-s + (−0.986 + 1.70i)16-s + (−1.81 − 3.14i)17-s + ⋯
L(s)  = 1  + (0.367 + 0.211i)2-s + (0.462 − 0.801i)3-s + (−0.410 − 0.710i)4-s + 0.0875i·5-s + (0.339 − 0.196i)6-s + (0.883 − 0.510i)7-s − 0.771i·8-s + (0.0714 + 0.123i)9-s + (−0.0185 + 0.0321i)10-s + (0.150 + 0.0869i)11-s − 0.759·12-s + (−0.941 − 0.337i)13-s + 0.432·14-s + (0.0702 + 0.0405i)15-s + (−0.246 + 0.427i)16-s + (−0.439 − 0.761i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.190 + 0.981i$
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: \((2,\ 403,\ (\ :1/2),\ 0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39646 - 1.15152i\)
\(L(\frac12)\) \(\approx\) \(1.39646 - 1.15152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.39 + 1.21i)T \)
31 \( 1 - iT \)
good2 \( 1 + (-0.519 - 0.299i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.801 + 1.38i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 0.195iT - 5T^{2} \)
7 \( 1 + (-2.33 + 1.34i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.499 - 0.288i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.81 + 3.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.74 - 1.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 + 2.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.15 + 2.00i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (-6.65 - 3.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.10 + 4.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.73 - 4.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.19iT - 47T^{2} \)
53 \( 1 - 5.67T + 53T^{2} \)
59 \( 1 + (7.50 - 4.33i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.54 - 4.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.44 - 3.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.45 + 1.99i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.07iT - 73T^{2} \)
79 \( 1 - 2.16T + 79T^{2} \)
83 \( 1 + 6.83iT - 83T^{2} \)
89 \( 1 + (-11.6 - 6.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.7 + 6.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.95543233399909892765564813915, −10.23461163461336472210269601820, −9.168235394657934706423308478138, −8.119387848782755104380868538998, −7.27734103881211215993977880238, −6.46622547897884141921314191435, −5.03048844481583773922989366348, −4.44053758292793143765390761317, −2.56257017321510831456411048759, −1.13947383246798656290786338573, 2.28121427738528981144887937509, 3.54356239923650835747687544350, 4.50796504843666551099939683577, 5.17792063285645377141170718148, 6.83242242470184136785394177372, 8.082339231401628343844657282206, 8.786228502069953996608007535462, 9.437791672586386536196760080442, 10.61808428555820962979939002806, 11.56623911106981863533592766176

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