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Properties

Label 2-403-31.16-c1-0-29
Degree $2$
Conductor $403$
Sign $0.579 + 0.814i$
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  + (2.20 − 1.60i)2-s + (2.19 + 1.59i)3-s + (1.68 − 5.17i)4-s − 1.64·5-s + 7.38·6-s + (−0.619 + 1.90i)7-s + (−2.89 − 8.92i)8-s + (1.34 + 4.12i)9-s + (−3.61 + 2.62i)10-s + (−0.861 + 2.65i)11-s + (11.9 − 8.66i)12-s + (0.809 + 0.587i)13-s + (1.68 + 5.20i)14-s + (−3.59 − 2.61i)15-s + (−11.8 − 8.64i)16-s + (−2.01 − 6.21i)17-s + ⋯
L(s)  = 1  + (1.56 − 1.13i)2-s + (1.26 + 0.919i)3-s + (0.840 − 2.58i)4-s − 0.733·5-s + 3.01·6-s + (−0.234 + 0.720i)7-s + (−1.02 − 3.15i)8-s + (0.447 + 1.37i)9-s + (−1.14 + 0.831i)10-s + (−0.259 + 0.799i)11-s + (3.44 − 2.50i)12-s + (0.224 + 0.163i)13-s + (0.451 + 1.38i)14-s + (−0.928 − 0.674i)15-s + (−2.97 − 2.16i)16-s + (−0.489 − 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.579 + 0.814i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.579 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.37661 - 1.74115i\)
\(L(\frac12)\) \(\approx\) \(3.37661 - 1.74115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-3.16 - 4.58i)T \)
good2 \( 1 + (-2.20 + 1.60i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.19 - 1.59i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 1.64T + 5T^{2} \)
7 \( 1 + (0.619 - 1.90i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.861 - 2.65i)T + (-8.89 - 6.46i)T^{2} \)
17 \( 1 + (2.01 + 6.21i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.44 - 2.50i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.06 + 3.27i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.11 - 1.53i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 5.52T + 37T^{2} \)
41 \( 1 + (-9.45 + 6.86i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (2.54 - 1.85i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-8.20 - 5.95i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.0864 - 0.265i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.10 - 1.52i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 0.832T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 + (-3.81 - 11.7i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.829 + 2.55i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.18 + 3.64i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (9.20 - 6.68i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.80 + 11.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.10 + 15.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.26334896850674594175165519688, −10.35517484679669485075941984223, −9.570876196552879172023993553668, −8.772915083442993879196946504717, −7.30446089770854178980198050640, −5.86557774940853686155492524766, −4.55174912539403677668410941181, −4.15879278461853966942347152796, −2.95882477287016072045646355286, −2.30437688828272086788155757968, 2.49451900542159142452692781528, 3.67443945732829628617271647304, 4.16814427011458666639315839905, 5.91989138309511707525966570080, 6.67617040649631713862472078807, 7.77100290675042306450180195774, 7.947501783912077832304976272213, 8.898254626373029023333288120934, 10.88725495498632755566395417705, 11.87576930705695379507400324680

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