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Properties

Label 2-403-31.2-c1-0-17
Degree $2$
Conductor $403$
Sign $-0.993 - 0.112i$
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  + (−1.43 − 1.04i)2-s + (−1.43 + 1.03i)3-s + (0.358 + 1.10i)4-s + 0.805·5-s + 3.14·6-s + (−0.358 − 1.10i)7-s + (−0.461 + 1.42i)8-s + (0.0391 − 0.120i)9-s + (−1.15 − 0.841i)10-s + (−0.157 − 0.483i)11-s + (−1.65 − 1.20i)12-s + (0.809 − 0.587i)13-s + (−0.637 + 1.96i)14-s + (−1.15 + 0.837i)15-s + (4.02 − 2.92i)16-s + (1.68 − 5.19i)17-s + ⋯
L(s)  = 1  + (−1.01 − 0.738i)2-s + (−0.825 + 0.600i)3-s + (0.179 + 0.551i)4-s + 0.360·5-s + 1.28·6-s + (−0.135 − 0.417i)7-s + (−0.163 + 0.502i)8-s + (0.0130 − 0.0401i)9-s + (−0.366 − 0.266i)10-s + (−0.0473 − 0.145i)11-s + (−0.479 − 0.348i)12-s + (0.224 − 0.163i)13-s + (−0.170 + 0.524i)14-s + (−0.297 + 0.216i)15-s + (1.00 − 0.730i)16-s + (0.409 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.993 - 0.112i$
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: \((2,\ 403,\ (\ :1/2),\ -0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00692612 + 0.122808i\)
\(L(\frac12)\) \(\approx\) \(0.00692612 + 0.122808i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.54 + 0.543i)T \)
good2 \( 1 + (1.43 + 1.04i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (1.43 - 1.03i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 - 0.805T + 5T^{2} \)
7 \( 1 + (0.358 + 1.10i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.157 + 0.483i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (-1.68 + 5.19i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.28 + 2.38i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.80 - 8.64i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.281 + 0.204i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 7.84T + 37T^{2} \)
41 \( 1 + (4.00 + 2.90i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (6.59 + 4.79i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (8.33 - 6.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.71 - 5.26i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.6 + 7.73i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 7.23T + 61T^{2} \)
67 \( 1 + 9.14T + 67T^{2} \)
71 \( 1 + (-2.69 + 8.29i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.72 + 5.30i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.85 - 5.71i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.54 - 4.02i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.86 + 5.72i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.50 + 13.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.69860187249034182398156055952, −9.946469952340051588437211868000, −9.413609240117465351014560110483, −8.288402904546749367515253413365, −7.20525357942446151517682621486, −5.74532751394993672065868349045, −5.08674556875510776559036617106, −3.51096214667062752747829824750, −1.91323510998316582717430240743, −0.12120980197535065743719186559, 1.70823663435480742975989287623, 3.79780630115321088417657471792, 5.58643195235310354269895685938, 6.32301267663452276689897431847, 6.89316670378551260430295205488, 8.192011851677558474932657019366, 8.696930126131228284281715598342, 9.907130976756257164624443607661, 10.52895319578615633070335683802, 11.79964902399602830396898741711

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