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Properties

Label 2-403-13.9-c1-0-34
Degree $2$
Conductor $403$
Sign $-0.984 + 0.177i$
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  + (1.12 − 1.95i)2-s + (1.66 − 2.88i)3-s + (−1.53 − 2.66i)4-s − 0.820·5-s + (−3.74 − 6.48i)6-s + (1.96 + 3.40i)7-s − 2.41·8-s + (−4.03 − 6.98i)9-s + (−0.924 + 1.60i)10-s + (−1.09 + 1.89i)11-s − 10.2·12-s + (−0.276 + 3.59i)13-s + 8.86·14-s + (−1.36 + 2.36i)15-s + (0.353 − 0.612i)16-s + (2.05 + 3.56i)17-s + ⋯
L(s)  = 1  + (0.796 − 1.37i)2-s + (0.960 − 1.66i)3-s + (−0.767 − 1.33i)4-s − 0.366·5-s + (−1.52 − 2.64i)6-s + (0.744 + 1.28i)7-s − 0.853·8-s + (−1.34 − 2.32i)9-s + (−0.292 + 0.506i)10-s + (−0.329 + 0.570i)11-s − 2.94·12-s + (−0.0767 + 0.997i)13-s + 2.36·14-s + (−0.352 + 0.610i)15-s + (0.0884 − 0.153i)16-s + (0.499 + 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.984 + 0.177i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.232870 - 2.60115i\)
\(L(\frac12)\) \(\approx\) \(0.232870 - 2.60115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.276 - 3.59i)T \)
31 \( 1 - T \)
good2 \( 1 + (-1.12 + 1.95i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.66 + 2.88i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.820T + 5T^{2} \)
7 \( 1 + (-1.96 - 3.40i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.09 - 1.89i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.05 - 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.657 - 1.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.70 + 8.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.172 - 0.299i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (3.01 - 5.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.55 - 7.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.21 + 5.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.68T + 47T^{2} \)
53 \( 1 - 2.16T + 53T^{2} \)
59 \( 1 + (-2.19 - 3.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.59 - 4.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.25 + 2.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.68 + 2.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.64T + 73T^{2} \)
79 \( 1 + 7.24T + 79T^{2} \)
83 \( 1 + 1.24T + 83T^{2} \)
89 \( 1 + (-8.01 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.36 + 2.36i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.53066658377586655512556242677, −10.06238469556593517036788930048, −8.815757379589034194326749164411, −8.257338991833296210558935307946, −7.15218999735216860260741902685, −5.96379262978573787089469260906, −4.63064690032407372258122957375, −3.22601809939770862860670071360, −2.25392730998943166074933424928, −1.58193755148591024770348267366, 3.30189924661434023488032621859, 3.89416682988812884328711426157, 5.00251748887967663195577376145, 5.41408168986509825114539970254, 7.40757961306831870712718097138, 7.77412151523759154471137185571, 8.645966992145569638545185482330, 9.803520516695036593945158848529, 10.65836567439114284448426809255, 11.42046893342458331204560916096

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