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Properties

Label 2-403-31.4-c1-0-1
Degree $2$
Conductor $403$
Sign $-0.725 + 0.687i$
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  + (0.773 + 2.37i)2-s + (0.384 − 1.18i)3-s + (−3.44 + 2.50i)4-s − 2.86·5-s + 3.11·6-s + (−1.19 + 0.866i)7-s + (−4.56 − 3.31i)8-s + (1.17 + 0.853i)9-s + (−2.21 − 6.81i)10-s + (−4.28 + 3.11i)11-s + (1.63 + 5.03i)12-s + (0.309 − 0.951i)13-s + (−2.98 − 2.16i)14-s + (−1.10 + 3.38i)15-s + (1.73 − 5.33i)16-s + (−6.22 − 4.52i)17-s + ⋯
L(s)  = 1  + (0.546 + 1.68i)2-s + (0.221 − 0.683i)3-s + (−1.72 + 1.25i)4-s − 1.28·5-s + 1.27·6-s + (−0.451 + 0.327i)7-s + (−1.61 − 1.17i)8-s + (0.391 + 0.284i)9-s + (−0.699 − 2.15i)10-s + (−1.29 + 0.938i)11-s + (0.472 + 1.45i)12-s + (0.0857 − 0.263i)13-s + (−0.797 − 0.579i)14-s + (−0.284 + 0.874i)15-s + (0.433 − 1.33i)16-s + (−1.51 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.725 + 0.687i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.725 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263697 - 0.661716i\)
\(L(\frac12)\) \(\approx\) \(0.263697 - 0.661716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-3.97 - 3.89i)T \)
good2 \( 1 + (-0.773 - 2.37i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.384 + 1.18i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + 2.86T + 5T^{2} \)
7 \( 1 + (1.19 - 0.866i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (4.28 - 3.11i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (6.22 + 4.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.422 - 1.30i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.11 - 2.26i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.139 + 0.430i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 5.08T + 37T^{2} \)
41 \( 1 + (-3.74 - 11.5i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.47 - 7.62i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (1.09 - 3.36i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.503 + 0.365i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.44 + 4.43i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 - 6.12T + 67T^{2} \)
71 \( 1 + (6.31 + 4.59i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.48 - 2.53i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-7.59 - 5.51i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.95 + 15.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-3.75 + 2.73i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (6.67 - 4.85i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.30102712375825918014307221505, −11.10889222040812645819964182100, −9.656066213732383433780802312245, −8.451723464458032986146748576345, −7.76757436012114996427580322441, −7.23519187358752483767100515566, −6.49252969958617604767242110965, −5.02143704480778916291978250460, −4.43739652013916952100109849203, −2.86518884658332195207034832919, 0.37180178598830565788711981871, 2.56241967476865268865711297069, 3.76121455027205658848174340332, 4.07040831322645047379807999083, 5.23083226379602472031842874699, 6.90352049941448856744603391904, 8.387662846434546604378251970041, 9.088623889906853216211390548258, 10.35716700515740522946820917440, 10.69684133453139173342087485457

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