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Properties

Label 2-403-13.12-c1-0-6
Degree $2$
Conductor $403$
Sign $0.591 + 0.806i$
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  − 2.21i·2-s − 2.59·3-s − 2.91·4-s + 1.00i·5-s + 5.74i·6-s + 1.82i·7-s + 2.02i·8-s + 3.71·9-s + 2.22·10-s + 2.73i·11-s + 7.55·12-s + (2.90 − 2.13i)13-s + 4.03·14-s − 2.60i·15-s − 1.33·16-s + 2.66·17-s + ⋯
L(s)  = 1  − 1.56i·2-s − 1.49·3-s − 1.45·4-s + 0.449i·5-s + 2.34i·6-s + 0.688i·7-s + 0.716i·8-s + 1.23·9-s + 0.705·10-s + 0.824i·11-s + 2.18·12-s + (0.806 − 0.591i)13-s + 1.07·14-s − 0.673i·15-s − 0.334·16-s + 0.647·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.591 + 0.806i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.591 + 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670120 - 0.339623i\)
\(L(\frac12)\) \(\approx\) \(0.670120 - 0.339623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.90 + 2.13i)T \)
31 \( 1 + iT \)
good2 \( 1 + 2.21iT - 2T^{2} \)
3 \( 1 + 2.59T + 3T^{2} \)
5 \( 1 - 1.00iT - 5T^{2} \)
7 \( 1 - 1.82iT - 7T^{2} \)
11 \( 1 - 2.73iT - 11T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 - 2.04iT - 19T^{2} \)
23 \( 1 - 1.36T + 23T^{2} \)
29 \( 1 + 3.75T + 29T^{2} \)
37 \( 1 - 5.04iT - 37T^{2} \)
41 \( 1 + 3.02iT - 41T^{2} \)
43 \( 1 - 9.12T + 43T^{2} \)
47 \( 1 - 7.62iT - 47T^{2} \)
53 \( 1 - 5.26T + 53T^{2} \)
59 \( 1 - 2.66iT - 59T^{2} \)
61 \( 1 - 7.14T + 61T^{2} \)
67 \( 1 - 9.48iT - 67T^{2} \)
71 \( 1 - 4.99iT - 71T^{2} \)
73 \( 1 - 4.50iT - 73T^{2} \)
79 \( 1 + 0.813T + 79T^{2} \)
83 \( 1 + 1.18iT - 83T^{2} \)
89 \( 1 + 16.2iT - 89T^{2} \)
97 \( 1 + 3.69iT - 97T^{2} \)
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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

−11.18334458100207707168952020078, −10.50954244003541959547019269820, −9.894134163109172281528840818511, −8.771479240541126962395370318922, −7.24144093708370999440165267040, −6.06976899128578159639708529528, −5.21161746582651203037915500906, −4.01477904066591247004045890525, −2.69188731188820636254561960905, −1.16058977129461835611252743019, 0.75473608244935910638462840413, 4.00955705247480575518946125957, 5.08827481971264526280415277005, 5.75596208408127826239576749685, 6.57760959484391883205689000271, 7.31123291483485580078860569450, 8.436826751705589305169953329920, 9.310814124760996536490735449574, 10.72763518735831511912982314270, 11.22893866816759510091970665949

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