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Properties

Label 2-403-13.9-c1-0-4
Degree $2$
Conductor $403$
Sign $0.677 - 0.735i$
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  + (1.23 − 2.13i)2-s + (−1.38 + 2.39i)3-s + (−2.03 − 3.51i)4-s − 1.87·5-s + (3.40 + 5.89i)6-s + (1.18 + 2.05i)7-s − 5.07·8-s + (−2.32 − 4.02i)9-s + (−2.30 + 3.99i)10-s + (−2.52 + 4.37i)11-s + 11.2·12-s + (1.94 + 3.03i)13-s + 5.85·14-s + (2.59 − 4.48i)15-s + (−2.18 + 3.78i)16-s + (1.54 + 2.68i)17-s + ⋯
L(s)  = 1  + (0.870 − 1.50i)2-s + (−0.798 + 1.38i)3-s + (−1.01 − 1.75i)4-s − 0.837·5-s + (1.39 + 2.40i)6-s + (0.449 + 0.778i)7-s − 1.79·8-s + (−0.775 − 1.34i)9-s + (−0.728 + 1.26i)10-s + (−0.761 + 1.31i)11-s + 3.24·12-s + (0.540 + 0.841i)13-s + 1.56·14-s + (0.668 − 1.15i)15-s + (−0.546 + 0.947i)16-s + (0.375 + 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.677 - 0.735i$
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: \((2,\ 403,\ (\ :1/2),\ 0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951123 + 0.417173i\)
\(L(\frac12)\) \(\approx\) \(0.951123 + 0.417173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.94 - 3.03i)T \)
31 \( 1 + T \)
good2 \( 1 + (-1.23 + 2.13i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.38 - 2.39i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.87T + 5T^{2} \)
7 \( 1 + (-1.18 - 2.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.52 - 4.37i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.54 - 2.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.102 - 0.177i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.80 - 6.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.41 + 5.92i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (2.50 - 4.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.20 + 10.7i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.35 - 4.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.735T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + (-2.19 - 3.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.718 - 1.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0449 + 0.0779i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.60 - 2.78i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.37T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 5.37T + 83T^{2} \)
89 \( 1 + (1.90 - 3.30i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.90 - 8.49i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.55995930651390994976174149888, −10.70521983440311382429073187332, −9.967754378560084511775882055397, −9.286169424017868832236387273500, −7.85275258708872529432947335328, −5.91874013580196839685087218001, −5.08019805486505620094854968110, −4.28822717843542634015552152137, −3.66343531246843155662303689299, −2.05140155141786883235324185560, 0.57205215582552374237340781738, 3.31254193009443743989084288909, 4.66669230410315230686894207829, 5.67185714343737810793870396858, 6.35769585998830568081987506148, 7.34052251131569301119906811435, 7.927148521765798690193644026113, 8.384313603093802915997489884590, 10.69846808720095060117215324743, 11.36022267981868767790433652294

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