Properties

Label 2-403-403.16-c1-0-23
Degree $2$
Conductor $403$
Sign $-0.0606 + 0.998i$
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  + (0.376 + 0.167i)2-s + (−0.276 + 2.63i)3-s + (−1.22 − 1.35i)4-s − 2.68·5-s + (−0.546 + 0.945i)6-s + (−0.664 − 0.738i)7-s + (−0.488 − 1.50i)8-s + (−3.92 − 0.833i)9-s + (−1.01 − 0.450i)10-s + (2.76 − 0.588i)11-s + (3.91 − 2.84i)12-s + (1.90 − 3.06i)13-s + (−0.126 − 0.389i)14-s + (0.743 − 7.07i)15-s + (−0.314 + 2.99i)16-s + (−6.04 − 1.28i)17-s + ⋯
L(s)  = 1  + (0.266 + 0.118i)2-s + (−0.159 + 1.52i)3-s + (−0.612 − 0.679i)4-s − 1.20·5-s + (−0.222 + 0.386i)6-s + (−0.251 − 0.279i)7-s + (−0.172 − 0.531i)8-s + (−1.30 − 0.277i)9-s + (−0.320 − 0.142i)10-s + (0.834 − 0.177i)11-s + (1.13 − 0.822i)12-s + (0.527 − 0.849i)13-s + (−0.0338 − 0.104i)14-s + (0.191 − 1.82i)15-s + (−0.0786 + 0.747i)16-s + (−1.46 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0606 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0606 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.0606 + 0.998i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.0606 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255453 - 0.271452i\)
\(L(\frac12)\) \(\approx\) \(0.255453 - 0.271452i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.90 + 3.06i)T \)
31 \( 1 + (-0.595 - 5.53i)T \)
good2 \( 1 + (-0.376 - 0.167i)T + (1.33 + 1.48i)T^{2} \)
3 \( 1 + (0.276 - 2.63i)T + (-2.93 - 0.623i)T^{2} \)
5 \( 1 + 2.68T + 5T^{2} \)
7 \( 1 + (0.664 + 0.738i)T + (-0.731 + 6.96i)T^{2} \)
11 \( 1 + (-2.76 + 0.588i)T + (10.0 - 4.47i)T^{2} \)
17 \( 1 + (6.04 + 1.28i)T + (15.5 + 6.91i)T^{2} \)
19 \( 1 + (0.535 + 5.09i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-1.46 + 1.62i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (9.26 + 4.12i)T + (19.4 + 21.5i)T^{2} \)
37 \( 1 + (1.98 + 3.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.657 - 0.292i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (0.103 + 0.983i)T + (-42.0 + 8.94i)T^{2} \)
47 \( 1 + (1.89 + 1.37i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.32 - 7.16i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.99 - 3.55i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (1.92 - 3.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.577 + 1.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.68 + 0.571i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (1.06 - 3.28i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.34 + 13.3i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.880 - 0.639i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (10.8 - 2.30i)T + (81.3 - 36.1i)T^{2} \)
97 \( 1 + (-8.48 - 9.41i)T + (-10.1 + 96.4i)T^{2} \)
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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

−10.96813779970502485212294602797, −10.23072212669012097121441341421, −9.077383967977516995580432644955, −8.798200292471143005419304705664, −7.18264242355782267152060119129, −5.98213089107371455600125780549, −4.81713726184444445736028080393, −4.17718295714302418339587850252, −3.43799606872890713073428608557, −0.23077660683493870592107097242, 1.84809495542238410878880200634, 3.54308222391528513008248259936, 4.35107552805541240332711571503, 6.01574920849538895539747075979, 6.95760160743986507744301330864, 7.73302538660354745492714327891, 8.527524301473896740039486324213, 9.294315770337155092231921003077, 11.34429545760920248365602832990, 11.57350302121498068926184476808

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