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Properties

Label 2-403-13.12-c1-0-33
Degree $2$
Conductor $403$
Sign $0.312 - 0.949i$
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  − 2.67i·2-s − 1.67·3-s − 5.15·4-s − 2.86i·5-s + 4.46i·6-s − 5.21i·7-s + 8.44i·8-s − 0.209·9-s − 7.66·10-s + 0.107i·11-s + 8.61·12-s + (3.42 + 1.12i)13-s − 13.9·14-s + 4.78i·15-s + 12.2·16-s − 1.65·17-s + ⋯
L(s)  = 1  − 1.89i·2-s − 0.964·3-s − 2.57·4-s − 1.28i·5-s + 1.82i·6-s − 1.96i·7-s + 2.98i·8-s − 0.0698·9-s − 2.42·10-s + 0.0323i·11-s + 2.48·12-s + (0.949 + 0.312i)13-s − 3.72·14-s + 1.23i·15-s + 3.06·16-s − 0.401·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.312 - 0.949i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.312 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.514065 + 0.372139i\)
\(L(\frac12)\) \(\approx\) \(0.514065 + 0.372139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.42 - 1.12i)T \)
31 \( 1 - iT \)
good2 \( 1 + 2.67iT - 2T^{2} \)
3 \( 1 + 1.67T + 3T^{2} \)
5 \( 1 + 2.86iT - 5T^{2} \)
7 \( 1 + 5.21iT - 7T^{2} \)
11 \( 1 - 0.107iT - 11T^{2} \)
17 \( 1 + 1.65T + 17T^{2} \)
19 \( 1 + 2.74iT - 19T^{2} \)
23 \( 1 - 2.05T + 23T^{2} \)
29 \( 1 - 8.54T + 29T^{2} \)
37 \( 1 - 7.15iT - 37T^{2} \)
41 \( 1 + 9.28iT - 41T^{2} \)
43 \( 1 + 7.32T + 43T^{2} \)
47 \( 1 - 4.70iT - 47T^{2} \)
53 \( 1 + 3.04T + 53T^{2} \)
59 \( 1 + 11.4iT - 59T^{2} \)
61 \( 1 + 5.46T + 61T^{2} \)
67 \( 1 + 1.60iT - 67T^{2} \)
71 \( 1 + 8.50iT - 71T^{2} \)
73 \( 1 - 6.98iT - 73T^{2} \)
79 \( 1 + 1.90T + 79T^{2} \)
83 \( 1 + 9.64iT - 83T^{2} \)
89 \( 1 - 9.70iT - 89T^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−10.77314546453488965655898301064, −10.07938931054557473299832694617, −9.001925050725667133361829066979, −8.228864947728068757936702329040, −6.57022403713356969113180975783, −4.90227262001598199621769166222, −4.55975783314967675625157309752, −3.41636591831270242939291857552, −1.31477744630168698078845831534, −0.54772789827394013181865708734, 3.06189212499361988760764239449, 4.83217454028895991535604064226, 5.89814862460506081247698181508, 6.08564227730464750959918551555, 6.94254678416154925795155561372, 8.279550299972380336111718512902, 8.792450475129345069441919249709, 9.992554381831783206227288066930, 11.10334853868626082876066501791, 12.00835959945236987752287788093

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