Properties

Label 2-403-31.4-c1-0-21
Degree $2$
Conductor $403$
Sign $-0.338 + 0.940i$
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  + (−0.471 − 1.44i)2-s + (0.0410 − 0.126i)3-s + (−0.261 + 0.190i)4-s + 1.10·5-s − 0.202·6-s + (0.346 − 0.252i)7-s + (−2.06 − 1.50i)8-s + (2.41 + 1.75i)9-s + (−0.520 − 1.60i)10-s + (2.52 − 1.83i)11-s + (0.0132 + 0.0408i)12-s + (0.309 − 0.951i)13-s + (−0.528 − 0.384i)14-s + (0.0453 − 0.139i)15-s + (−1.40 + 4.32i)16-s + (0.591 + 0.429i)17-s + ⋯
L(s)  = 1  + (−0.333 − 1.02i)2-s + (0.0236 − 0.0729i)3-s + (−0.130 + 0.0950i)4-s + 0.494·5-s − 0.0826·6-s + (0.131 − 0.0952i)7-s + (−0.730 − 0.531i)8-s + (0.804 + 0.584i)9-s + (−0.164 − 0.507i)10-s + (0.761 − 0.553i)11-s + (0.00383 + 0.0117i)12-s + (0.0857 − 0.263i)13-s + (−0.141 − 0.102i)14-s + (0.0117 − 0.0360i)15-s + (−0.350 + 1.08i)16-s + (0.143 + 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.338 + 0.940i$
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: \((2,\ 403,\ (\ :1/2),\ -0.338 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809453 - 1.15195i\)
\(L(\frac12)\) \(\approx\) \(0.809453 - 1.15195i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.30 - 1.68i)T \)
good2 \( 1 + (0.471 + 1.44i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.0410 + 0.126i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 1.10T + 5T^{2} \)
7 \( 1 + (-0.346 + 0.252i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-2.52 + 1.83i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (-0.591 - 0.429i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.37 + 7.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.52 - 3.29i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.261 + 0.803i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 7.85T + 37T^{2} \)
41 \( 1 + (0.826 + 2.54i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.99 + 9.22i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.00 - 9.25i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.0102 + 0.00746i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.627 + 1.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 0.398T + 61T^{2} \)
67 \( 1 - 6.84T + 67T^{2} \)
71 \( 1 + (-11.9 - 8.70i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (10.7 - 7.82i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.72 - 1.98i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.26 - 10.0i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-14.2 + 10.3i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-6.29 + 4.57i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.96882894913012026798545150630, −10.20967732968594182049684284296, −9.385198662308980151476785769458, −8.586441020423378358109916682901, −7.17257646115796984258064909880, −6.32115715439175119548892041095, −5.00769084179405290681305265181, −3.64071119631429325209593740552, −2.35988751784420841935689201420, −1.17023922995861903244254738408, 1.81854881945625930532957851325, 3.61547664814925463957028429442, 4.94366325287624749855619576467, 6.26447137066946942424936724094, 6.68404330095793052977506059608, 7.77001598402534594114638219885, 8.690895658895850589365702130173, 9.573957910051612220818277293458, 10.29877597807338322349938368953, 11.74455037130032451572882647065

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