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Properties

Label 2-403-403.25-c1-0-35
Degree $2$
Conductor $403$
Sign $0.567 - 0.823i$
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  − 2.00i·2-s + (−1.67 − 2.90i)3-s − 2.03·4-s + (−0.151 − 0.0877i)5-s + (−5.82 + 3.36i)6-s + (−2.39 + 1.38i)7-s + 0.0771i·8-s + (−4.10 + 7.11i)9-s + (−0.176 + 0.305i)10-s + (3.13 + 1.81i)11-s + (3.41 + 5.91i)12-s + (2.94 − 2.07i)13-s + (2.77 + 4.80i)14-s + 0.587i·15-s − 3.92·16-s + (−2.82 − 4.88i)17-s + ⋯
L(s)  = 1  − 1.42i·2-s + (−0.966 − 1.67i)3-s − 1.01·4-s + (−0.0679 − 0.0392i)5-s + (−2.37 + 1.37i)6-s + (−0.904 + 0.522i)7-s + 0.0272i·8-s + (−1.36 + 2.37i)9-s + (−0.0557 + 0.0965i)10-s + (0.946 + 0.546i)11-s + (0.985 + 1.70i)12-s + (0.817 − 0.575i)13-s + (0.741 + 1.28i)14-s + 0.151i·15-s − 0.980·16-s + (−0.684 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.567 - 0.823i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.347394 + 0.182572i\)
\(L(\frac12)\) \(\approx\) \(0.347394 + 0.182572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.94 + 2.07i)T \)
31 \( 1 + (2.55 + 4.94i)T \)
good2 \( 1 + 2.00iT - 2T^{2} \)
3 \( 1 + (1.67 + 2.90i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.151 + 0.0877i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.39 - 1.38i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.13 - 1.81i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.82 + 4.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.40 - 2.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.59T + 23T^{2} \)
29 \( 1 + 0.682T + 29T^{2} \)
37 \( 1 + (0.536 - 0.309i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.51 - 5.49i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 + 1.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.66iT - 47T^{2} \)
53 \( 1 + (-0.973 + 1.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.24 + 1.87i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.97T + 61T^{2} \)
67 \( 1 + (10.5 + 6.07i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (13.0 + 7.54i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.229 - 0.132i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.904 + 1.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.00 + 5.19i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.17iT - 89T^{2} \)
97 \( 1 + 7.88iT - 97T^{2} \)
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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

−10.91208133038140247604790737278, −9.874556492817642974011250158801, −8.810695496605322044295587770525, −7.58691642761274565630708655013, −6.44742624036824714106810518732, −6.00391340512798884523522955165, −4.29070363397811494831441815322, −2.69003736022503928741818698621, −1.72498335888586642111130263148, −0.28184407353435945981347962391, 3.82079971356687338231191052222, 4.20294258940879243694412217996, 5.64954347959594585488597927240, 6.24498746084391705718401438248, 6.86589952855709899802862616912, 8.705244223420148321429628445257, 9.032843576241585600917543906617, 10.19150235653535736843795607285, 10.99294466830814913787462262389, 11.65129282471943334139791513897

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