↑

Properties

Label 2-403-13.9-c1-0-0
Degree $2$
Conductor $403$
Sign $-0.359 + 0.933i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0573 + 0.0993i)2-s + (−1.57 + 2.72i)3-s + (0.993 + 1.72i)4-s − 1.85·5-s + (−0.180 − 0.312i)6-s + (−1.95 − 3.39i)7-s − 0.457·8-s + (−3.45 − 5.98i)9-s + (0.106 − 0.183i)10-s + (−1.20 + 2.07i)11-s − 6.25·12-s + (2.92 + 2.10i)13-s + 0.449·14-s + (2.91 − 5.04i)15-s + (−1.96 + 3.39i)16-s + (0.578 + 1.00i)17-s + ⋯
L(s)  = 1  + (−0.0405 + 0.0702i)2-s + (−0.908 + 1.57i)3-s + (0.496 + 0.860i)4-s − 0.827·5-s + (−0.0736 − 0.127i)6-s + (−0.740 − 1.28i)7-s − 0.161·8-s + (−1.15 − 1.99i)9-s + (0.0335 − 0.0581i)10-s + (−0.361 + 0.626i)11-s − 1.80·12-s + (0.811 + 0.583i)13-s + 0.120·14-s + (0.751 − 1.30i)15-s + (−0.490 + 0.848i)16-s + (0.140 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.359 + 0.933i$
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: \((2,\ 403,\ (\ :1/2),\ -0.359 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104816 - 0.152766i\)
\(L(\frac12)\) \(\approx\) \(0.104816 - 0.152766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.92 - 2.10i)T \)
31 \( 1 - T \)
good2 \( 1 + (0.0573 - 0.0993i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.57 - 2.72i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.85T + 5T^{2} \)
7 \( 1 + (1.95 + 3.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.20 - 2.07i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.578 - 1.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.84 + 4.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.14 + 1.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.80 - 3.12i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (-2.03 + 3.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.19 - 8.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.88 + 6.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.34T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + (2.25 + 3.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.15 - 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.01 + 5.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.806 - 1.39i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.223T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + (6.55 - 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.26 + 7.38i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.48558927915429780657621381735, −11.04535799661368754034548243908, −10.29869547044889376147980768475, −9.351465934870955194706899159254, −8.263214867375414313523705147932, −7.02930983209680859781174296788, −6.38400534123101376075633571767, −4.71524145807577603665684589717, −4.00633638984325472426826501103, −3.31735957919099681592030070655, 0.13105901060020775291930710658, 1.71513026761039142264023948192, 3.06201039381090685023949062388, 5.36173805384280001538164903662, 6.00114637569393081099439428557, 6.54200861275042402814347106588, 7.77261394916227502720318155498, 8.465453706372981683030172987098, 9.924582667423917829354441819358, 11.09843953065998080301338405416

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