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Properties

Label 2-403-13.3-c1-0-2
Degree $2$
Conductor $403$
Sign $-0.685 - 0.727i$
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  + (0.948 + 1.64i)2-s + (−1.29 − 2.23i)3-s + (−0.799 + 1.38i)4-s − 2.49·5-s + (2.45 − 4.24i)6-s + (−1.48 + 2.57i)7-s + 0.760·8-s + (−1.83 + 3.18i)9-s + (−2.36 − 4.09i)10-s + (2.82 + 4.89i)11-s + 4.13·12-s + (−1.91 + 3.05i)13-s − 5.63·14-s + (3.21 + 5.57i)15-s + (2.32 + 4.01i)16-s + (−1.64 + 2.85i)17-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)2-s + (−0.746 − 1.29i)3-s + (−0.399 + 0.692i)4-s − 1.11·5-s + (1.00 − 1.73i)6-s + (−0.561 + 0.971i)7-s + 0.268·8-s + (−0.613 + 1.06i)9-s + (−0.747 − 1.29i)10-s + (0.851 + 1.47i)11-s + 1.19·12-s + (−0.530 + 0.847i)13-s − 1.50·14-s + (0.831 + 1.43i)15-s + (0.580 + 1.00i)16-s + (−0.399 + 0.691i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.685 - 0.727i$
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: \((2,\ 403,\ (\ :1/2),\ -0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366666 + 0.849631i\)
\(L(\frac12)\) \(\approx\) \(0.366666 + 0.849631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.91 - 3.05i)T \)
31 \( 1 + T \)
good2 \( 1 + (-0.948 - 1.64i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.29 + 2.23i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.49T + 5T^{2} \)
7 \( 1 + (1.48 - 2.57i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.82 - 4.89i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.64 - 2.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.78 + 3.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.01 + 1.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.03 - 1.78i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (3.66 + 6.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.556 + 0.964i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.70 - 8.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.52T + 47T^{2} \)
53 \( 1 - 4.28T + 53T^{2} \)
59 \( 1 + (-1.00 + 1.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.31 + 2.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.55 - 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.22 + 9.05i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + 4.67T + 83T^{2} \)
89 \( 1 + (-1.64 - 2.84i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.66 - 2.88i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.03328762630161626178213128120, −11.17921433126530351786899089157, −9.572868935001070706627291428575, −8.380040064405083231086535177982, −7.30203809188364767137328227872, −6.89244336305790820048433907259, −6.21345867442225756042938251658, −5.04895699658085077054783684746, −4.06134435290937897489739303309, −1.94160724813421976986476082438, 0.53803055369812959640626514249, 3.40765757605923386844391285066, 3.63756038924475281343953411908, 4.60801062597498193272091960657, 5.63017932958120538653239009089, 7.09091426369485310565227221358, 8.291414877509805287604902934700, 9.751296303664205235101020086499, 10.28102108456119865897640496419, 11.16666100106470008118244033360

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