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Properties

Label 2-403-31.4-c1-0-10
Degree $2$
Conductor $403$
Sign $-0.319 - 0.947i$
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  + (0.767 + 2.36i)2-s + (0.517 − 1.59i)3-s + (−3.36 + 2.44i)4-s + 3.20·5-s + 4.16·6-s + (−2.60 + 1.89i)7-s + (−4.34 − 3.15i)8-s + (0.154 + 0.111i)9-s + (2.45 + 7.56i)10-s + (0.903 − 0.656i)11-s + (2.15 + 6.63i)12-s + (0.309 − 0.951i)13-s + (−6.45 − 4.69i)14-s + (1.65 − 5.10i)15-s + (1.54 − 4.76i)16-s + (4.43 + 3.22i)17-s + ⋯
L(s)  = 1  + (0.542 + 1.66i)2-s + (0.299 − 0.920i)3-s + (−1.68 + 1.22i)4-s + 1.43·5-s + 1.69·6-s + (−0.983 + 0.714i)7-s + (−1.53 − 1.11i)8-s + (0.0513 + 0.0373i)9-s + (0.776 + 2.39i)10-s + (0.272 − 0.197i)11-s + (0.622 + 1.91i)12-s + (0.0857 − 0.263i)13-s + (−1.72 − 1.25i)14-s + (0.428 − 1.31i)15-s + (0.386 − 1.19i)16-s + (1.07 + 0.782i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.319 - 0.947i$
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: \((2,\ 403,\ (\ :1/2),\ -0.319 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23487 + 1.71928i\)
\(L(\frac12)\) \(\approx\) \(1.23487 + 1.71928i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.20 - 1.97i)T \)
good2 \( 1 + (-0.767 - 2.36i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.517 + 1.59i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 3.20T + 5T^{2} \)
7 \( 1 + (2.60 - 1.89i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.903 + 0.656i)T + (3.39 - 10.4i)T^{2} \)
17 \( 1 + (-4.43 - 3.22i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.47 - 4.54i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.73 + 3.43i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.63 + 8.12i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 4.97T + 37T^{2} \)
41 \( 1 + (1.25 + 3.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.09 + 6.43i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.568 - 1.75i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.66 + 4.84i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.79 + 11.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 3.54T + 61T^{2} \)
67 \( 1 + 1.79T + 67T^{2} \)
71 \( 1 + (-1.28 - 0.935i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.0280 - 0.0204i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.78 + 3.47i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.45 - 10.6i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-10.7 + 7.81i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (12.7 - 9.28i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.24718167180994934861124123186, −10.17022218944951025961553031835, −9.480169386710007374882157616166, −8.381252981529945104913675264569, −7.68898521894660859892238489649, −6.47953027547063541211427748089, −6.06920186775583643761551801262, −5.41616871416729597572768341708, −3.67335407378796398669009520520, −2.04760793110234913181363564329, 1.38872980730159901671550731655, 2.88776556424991372978082726674, 3.65111306720079816470996379001, 4.73402268213266824237345939176, 5.72835649866590538646125613336, 7.05968860623746479664755579963, 9.299377859191153617976288425815, 9.475762102890286076345723232900, 10.05090241817611065190565310201, 10.77317281616764705122783194092

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