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Properties

Label 2-403-31.2-c1-0-22
Degree $2$
Conductor $403$
Sign $0.569 + 0.822i$
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  + (0.309 + 0.224i)2-s + (1.30 − 0.951i)3-s + (−0.572 − 1.76i)4-s + 2.61·5-s + 0.618·6-s + (−0.572 − 1.76i)7-s + (0.454 − 1.40i)8-s + (−0.118 + 0.363i)9-s + (0.809 + 0.587i)10-s + (1.30 + 4.02i)11-s + (−2.42 − 1.76i)12-s + (0.809 − 0.587i)13-s + (0.218 − 0.673i)14-s + (3.42 − 2.48i)15-s + (−2.54 + 1.84i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.158i)2-s + (0.755 − 0.549i)3-s + (−0.286 − 0.881i)4-s + 1.17·5-s + 0.252·6-s + (−0.216 − 0.666i)7-s + (0.160 − 0.495i)8-s + (−0.0393 + 0.121i)9-s + (0.255 + 0.185i)10-s + (0.394 + 1.21i)11-s + (−0.700 − 0.509i)12-s + (0.224 − 0.163i)13-s + (0.0584 − 0.180i)14-s + (0.884 − 0.642i)15-s + (−0.636 + 0.462i)16-s + (0.0749 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.569 + 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85796 - 0.973657i\)
\(L(\frac12)\) \(\approx\) \(1.85796 - 0.973657i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-3.23 - 4.53i)T \)
good2 \( 1 + (-0.309 - 0.224i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1.30 + 0.951i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 - 2.61T + 5T^{2} \)
7 \( 1 + (0.572 + 1.76i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.30 - 4.02i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.85 + 4.25i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.763 + 2.35i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.73 - 3.44i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 7.61T + 37T^{2} \)
41 \( 1 + (-2.92 - 2.12i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (2.73 + 1.98i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.16 + 3.02i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.64 - 11.2i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.11 + 0.812i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 + 5.61T + 67T^{2} \)
71 \( 1 + (2.85 - 8.78i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.16 + 3.57i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.545 - 1.67i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.61 + 3.35i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.16 - 3.57i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.70 - 11.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.66146102395254782786083835129, −10.26953834413430995691185341979, −9.276301804519266897171155582841, −8.560605032376576011941821859030, −7.01601797800836714868990197554, −6.62042417294740158187255403935, −5.29902243583887628143861419981, −4.35364655732982531019563745782, −2.51438683729114736769638287689, −1.45490197975037869442224897433, 2.24639199568136989083768910948, 3.28382901299105172966929209195, 4.17853702177236375577163524053, 5.70632261837651025176044983817, 6.42542116824553366840691222723, 8.207289021053130876874812335666, 8.687027125684970795815381162393, 9.417846775277348054868520741233, 10.29635612301668739870798353753, 11.51720957303242575966078416805

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