Properties

Label 2-403-13.3-c1-0-29
Degree $2$
Conductor $403$
Sign $0.761 + 0.647i$
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  + (−0.0496 − 0.0859i)2-s + (0.816 + 1.41i)3-s + (0.995 − 1.72i)4-s − 0.352·5-s + (0.0810 − 0.140i)6-s + (2.05 − 3.56i)7-s − 0.396·8-s + (0.167 − 0.289i)9-s + (0.0174 + 0.0302i)10-s + (−2.45 − 4.24i)11-s + 3.24·12-s + (−1.56 + 3.24i)13-s − 0.408·14-s + (−0.287 − 0.498i)15-s + (−1.97 − 3.41i)16-s + (−1.84 + 3.19i)17-s + ⋯
L(s)  = 1  + (−0.0350 − 0.0607i)2-s + (0.471 + 0.816i)3-s + (0.497 − 0.861i)4-s − 0.157·5-s + (0.0330 − 0.0572i)6-s + (0.777 − 1.34i)7-s − 0.140·8-s + (0.0557 − 0.0966i)9-s + (0.00552 + 0.00957i)10-s + (−0.739 − 1.28i)11-s + 0.937·12-s + (−0.434 + 0.900i)13-s − 0.109·14-s + (−0.0742 − 0.128i)15-s + (−0.492 − 0.853i)16-s + (−0.448 + 0.776i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.761 + 0.647i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.761 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62279 - 0.596747i\)
\(L(\frac12)\) \(\approx\) \(1.62279 - 0.596747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.56 - 3.24i)T \)
31 \( 1 - T \)
good2 \( 1 + (0.0496 + 0.0859i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.816 - 1.41i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.352T + 5T^{2} \)
7 \( 1 + (-2.05 + 3.56i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.45 + 4.24i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.84 - 3.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.16 + 3.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.83 - 4.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.66 - 6.35i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-4.65 - 8.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.57 - 6.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.204 + 0.353i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.39T + 47T^{2} \)
53 \( 1 + 7.56T + 53T^{2} \)
59 \( 1 + (-0.912 + 1.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.33 + 7.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.65 - 9.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.13 - 1.97i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.21T + 73T^{2} \)
79 \( 1 - 2.80T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + (6.24 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.37 + 11.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.12799397230981620721341931198, −10.27411776719685040052865534566, −9.559563931390758219029706713557, −8.514476011321520314906992114834, −7.42832962655128599751006003371, −6.48237393970431464958038961303, −5.11497416494853575069534417596, −4.25644730191737155743151864880, −3.02449012569668780827172160669, −1.20401585969865651689649260896, 2.25265900159394355573527191467, 2.57103529085094340622931925706, 4.50403294961697563537505470377, 5.64736157574041745957813551908, 7.04171084190781838437047531348, 7.84334324126607723037104389029, 8.138446936760966063801859905989, 9.325955390845199288733275117568, 10.56680861147544389997322544087, 11.68573223694051467914380830803

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