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Properties

Label 2-403-31.16-c1-0-12
Degree $2$
Conductor $403$
Sign $0.449 + 0.893i$
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  + (−0.0675 + 0.0491i)2-s + (−2.70 − 1.96i)3-s + (−0.615 + 1.89i)4-s + 0.141·5-s + 0.279·6-s + (−0.415 + 1.28i)7-s + (−0.103 − 0.317i)8-s + (2.53 + 7.79i)9-s + (−0.00958 + 0.00696i)10-s + (1.38 − 4.26i)11-s + (5.39 − 3.92i)12-s + (0.809 + 0.587i)13-s + (−0.0347 − 0.106i)14-s + (−0.383 − 0.278i)15-s + (−3.20 − 2.32i)16-s + (−0.105 − 0.323i)17-s + ⋯
L(s)  = 1  + (−0.0477 + 0.0347i)2-s + (−1.56 − 1.13i)3-s + (−0.307 + 0.947i)4-s + 0.0634·5-s + 0.114·6-s + (−0.157 + 0.483i)7-s + (−0.0364 − 0.112i)8-s + (0.844 + 2.59i)9-s + (−0.00303 + 0.00220i)10-s + (0.417 − 1.28i)11-s + (1.55 − 1.13i)12-s + (0.224 + 0.163i)13-s + (−0.00928 − 0.0285i)14-s + (−0.0991 − 0.0720i)15-s + (−0.800 − 0.581i)16-s + (−0.0255 − 0.0785i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.449 + 0.893i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.565574 - 0.348484i\)
\(L(\frac12)\) \(\approx\) \(0.565574 - 0.348484i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 - 5.00i)T \)
good2 \( 1 + (0.0675 - 0.0491i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.70 + 1.96i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 - 0.141T + 5T^{2} \)
7 \( 1 + (0.415 - 1.28i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.38 + 4.26i)T + (-8.89 - 6.46i)T^{2} \)
17 \( 1 + (0.105 + 0.323i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.50 + 3.27i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.96 + 6.04i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.40 + 2.47i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 5.02T + 37T^{2} \)
41 \( 1 + (-7.39 + 5.37i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-6.84 + 4.97i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (0.904 + 0.657i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.32 + 4.09i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-11.9 - 8.69i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 6.59T + 67T^{2} \)
71 \( 1 + (2.84 + 8.74i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.296 + 0.911i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.736 - 2.26i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.01 - 2.19i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.60 - 4.93i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-4.40 + 13.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.46900549081621823806636955212, −10.55857250007669623795349969606, −9.085262060845640994765397790918, −8.140267891740378596379651957881, −7.24619909652519721290272987196, −6.31722653443004838112837578093, −5.60128920703808963820664501602, −4.34473330146966206399330543123, −2.62098757035642221539649079954, −0.67136118499622300262630757875, 1.16305924432633778629186348431, 3.93675551517536237634528149452, 4.62475967338728150638614089442, 5.67008403296527964287335601424, 6.23248020531794695478965091536, 7.46354114008117465585425377303, 9.463204922447301679808271683093, 9.722385547318152751114805832899, 10.37445413144525158698946041550, 11.32639495621995614099554425345

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