Properties

Label 2-403-403.126-c1-0-11
Degree $2$
Conductor $403$
Sign $0.661 + 0.749i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.39 + 1.06i)2-s + (0.233 + 2.22i)3-s + (3.24 − 3.60i)4-s − 3·5-s + (−2.92 − 5.06i)6-s + (−2.00 + 2.22i)7-s + (−2.30 + 7.10i)8-s + (−1.95 + 0.415i)9-s + (7.17 − 3.19i)10-s + (−2.18 − 0.464i)11-s + (8.78 + 6.37i)12-s + (−0.495 + 3.57i)13-s + (2.42 − 7.46i)14-s + (−0.701 − 6.67i)15-s + (−1.03 − 9.80i)16-s + (1.35 − 0.287i)17-s + ⋯
L(s)  = 1  + (−1.69 + 0.752i)2-s + (0.134 + 1.28i)3-s + (1.62 − 1.80i)4-s − 1.34·5-s + (−1.19 − 2.06i)6-s + (−0.758 + 0.842i)7-s + (−0.816 + 2.51i)8-s + (−0.652 + 0.138i)9-s + (2.26 − 1.01i)10-s + (−0.659 − 0.140i)11-s + (2.53 + 1.84i)12-s + (−0.137 + 0.990i)13-s + (0.648 − 1.99i)14-s + (−0.181 − 1.72i)15-s + (−0.257 − 2.45i)16-s + (0.327 − 0.0696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.661 + 0.749i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.661 + 0.749i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.495 - 3.57i)T \)
31 \( 1 + (-5.28 - 1.76i)T \)
good2 \( 1 + (2.39 - 1.06i)T + (1.33 - 1.48i)T^{2} \)
3 \( 1 + (-0.233 - 2.22i)T + (-2.93 + 0.623i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (2.00 - 2.22i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (2.18 + 0.464i)T + (10.0 + 4.47i)T^{2} \)
17 \( 1 + (-1.35 + 0.287i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-0.627 + 5.96i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (2.51 + 2.79i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (5.13 - 2.28i)T + (19.4 - 21.5i)T^{2} \)
37 \( 1 + (-4.04 + 7.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.69 + 2.97i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (-0.676 + 6.43i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (6.85 - 4.97i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.118 - 0.363i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.17 - 2.75i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (5.78 + 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.11 - 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.42 + 0.941i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (2.14 + 6.60i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.07 - 15.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (9.66 + 7.02i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.81 - 0.385i)T + (81.3 + 36.1i)T^{2} \)
97 \( 1 + (-0.413 + 0.459i)T + (-10.1 - 96.4i)T^{2} \)
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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

−10.88447490928848840955800948041, −9.877584516171040411380671315122, −9.202954693893475756822278402616, −8.661444878247458708617066619014, −7.64292417457404277214545868698, −6.78296595461466753621309643550, −5.53409627016609236461521871133, −4.25325330144221261186770234007, −2.74594843916206594532864652849, 0, 1.19367856040792203542913487642, 2.80065530940836855957605674856, 3.79941532502813984928709922320, 6.30284885440535490760148922645, 7.47491859388600944546260055246, 7.78729894525651364041275610507, 8.206173169680364083073732446143, 9.803075276846769166036326730379, 10.28465143743651696197000987523

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