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Properties

Label 2-403-13.10-c1-0-10
Degree $2$
Conductor $403$
Sign $-0.764 - 0.644i$
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  + (−0.244 + 0.141i)2-s + (1.61 + 2.79i)3-s + (−0.960 + 1.66i)4-s − 0.0582i·5-s + (−0.789 − 0.455i)6-s + (2.44 + 1.41i)7-s − 1.10i·8-s + (−3.69 + 6.40i)9-s + (0.00822 + 0.0142i)10-s + (−1.55 + 0.899i)11-s − 6.18·12-s + (0.590 − 3.55i)13-s − 0.798·14-s + (0.162 − 0.0938i)15-s + (−1.76 − 3.05i)16-s + (3.07 − 5.33i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.0999i)2-s + (0.930 + 1.61i)3-s + (−0.480 + 0.831i)4-s − 0.0260i·5-s + (−0.322 − 0.186i)6-s + (0.925 + 0.534i)7-s − 0.391i·8-s + (−1.23 + 2.13i)9-s + (0.00260 + 0.00450i)10-s + (−0.469 + 0.271i)11-s − 1.78·12-s + (0.163 − 0.986i)13-s − 0.213·14-s + (0.0419 − 0.0242i)15-s + (−0.440 − 0.763i)16-s + (0.746 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.764 - 0.644i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.764 - 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543664 + 1.48747i\)
\(L(\frac12)\) \(\approx\) \(0.543664 + 1.48747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.590 + 3.55i)T \)
31 \( 1 - iT \)
good2 \( 1 + (0.244 - 0.141i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.61 - 2.79i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.0582iT - 5T^{2} \)
7 \( 1 + (-2.44 - 1.41i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.55 - 0.899i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.07 + 5.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.61 - 2.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.85 + 4.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.14 - 1.98i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (5.60 - 3.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.73 - 4.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.27 + 2.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.27iT - 47T^{2} \)
53 \( 1 + 8.65T + 53T^{2} \)
59 \( 1 + (-12.5 - 7.22i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.33 - 2.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.4 + 6.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.89 + 2.82i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.08iT - 73T^{2} \)
79 \( 1 + 1.74T + 79T^{2} \)
83 \( 1 + 5.82iT - 83T^{2} \)
89 \( 1 + (-6.18 + 3.57i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.26 - 4.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.51335892766802661125133419891, −10.31847649416914087361627885243, −9.801057491174806050849425143487, −8.714043414232670365962173399872, −8.296882033353078018124267581936, −7.49078739183891756790904011137, −5.17914693800946497378501970466, −4.87175486780231040941710425541, −3.48464172867722982151565809618, −2.76274502687005261432561089664, 1.13811149110151473061592355600, 1.97021645443718570834091070808, 3.63209102669310397403692760886, 5.21354742598375663663685248523, 6.34232761338929930971302269617, 7.35007390322695450449025189111, 8.155900958110534295109496030534, 8.822385208125797261719364064048, 9.841188072102445242522317226733, 11.02729292175966386045260807110

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