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Properties

Label 2-403-13.9-c1-0-11
Degree $2$
Conductor $403$
Sign $-0.977 + 0.211i$
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  + (−1.33 + 2.30i)2-s + (−1.12 + 1.94i)3-s + (−2.54 − 4.41i)4-s + 4.27·5-s + (−2.98 − 5.17i)6-s + (1.44 + 2.49i)7-s + 8.23·8-s + (−1.01 − 1.75i)9-s + (−5.69 + 9.85i)10-s + (0.929 − 1.60i)11-s + 11.4·12-s + (−0.150 + 3.60i)13-s − 7.66·14-s + (−4.79 + 8.30i)15-s + (−5.87 + 10.1i)16-s + (2.25 + 3.91i)17-s + ⋯
L(s)  = 1  + (−0.941 + 1.63i)2-s + (−0.647 + 1.12i)3-s + (−1.27 − 2.20i)4-s + 1.91·5-s + (−1.21 − 2.11i)6-s + (0.544 + 0.942i)7-s + 2.91·8-s + (−0.338 − 0.585i)9-s + (−1.79 + 3.11i)10-s + (0.280 − 0.485i)11-s + 3.29·12-s + (−0.0417 + 0.999i)13-s − 2.04·14-s + (−1.23 + 2.14i)15-s + (−1.46 + 2.54i)16-s + (0.547 + 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.977 + 0.211i$
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: \((2,\ 403,\ (\ :1/2),\ -0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101131 - 0.943320i\)
\(L(\frac12)\) \(\approx\) \(0.101131 - 0.943320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.150 - 3.60i)T \)
31 \( 1 - T \)
good2 \( 1 + (1.33 - 2.30i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.12 - 1.94i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 4.27T + 5T^{2} \)
7 \( 1 + (-1.44 - 2.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.929 + 1.60i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.25 - 3.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.62 + 2.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.44 + 2.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.29 - 2.23i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.00 + 6.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.408 + 0.706i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.93T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + (2.65 + 4.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.98 + 5.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.949 - 1.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.03 - 1.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.58T + 73T^{2} \)
79 \( 1 + 8.08T + 79T^{2} \)
83 \( 1 + 0.931T + 83T^{2} \)
89 \( 1 + (6.58 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.00 + 1.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.11611593034470118802538689436, −10.46551026819932886349294912080, −9.510985563835410383707855741258, −9.193488584237312881385894034749, −8.360378909541705543686950390382, −6.71785652194717316921023266490, −5.99630328521595753690694362372, −5.42903482847532788090288568684, −4.65882208558313574510718381007, −1.79900013989307492850154315569, 1.07191838415608519926323470205, 1.68210574878404104131617386371, 2.94232048514077564641062437060, 4.78030507239370518082451334888, 6.07009681257349794389624768196, 7.28531615486555059212549282760, 8.097814430931317432849930390036, 9.517010962279651655384603371464, 9.908473815949545805857798244579, 10.74683637891309478895200304144

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