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Properties

Label 2-403-31.2-c1-0-3
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: $\chi_{403} (157, \cdot )$
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

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

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