Properties

Label 2-403-13.9-c1-0-23
Degree $2$
Conductor $403$
Sign $0.994 + 0.102i$
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  + (−1.19 + 2.07i)2-s + (1.01 − 1.75i)3-s + (−1.86 − 3.22i)4-s + 3.38·5-s + (2.42 + 4.19i)6-s + (−2.00 − 3.47i)7-s + 4.12·8-s + (−0.553 − 0.958i)9-s + (−4.05 + 7.02i)10-s + (−1.52 + 2.64i)11-s − 7.54·12-s + (1.26 − 3.37i)13-s + 9.61·14-s + (3.43 − 5.94i)15-s + (−1.21 + 2.09i)16-s + (1.60 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.845 + 1.46i)2-s + (0.584 − 1.01i)3-s + (−0.931 − 1.61i)4-s + 1.51·5-s + (0.989 + 1.71i)6-s + (−0.759 − 1.31i)7-s + 1.45·8-s + (−0.184 − 0.319i)9-s + (−1.28 + 2.22i)10-s + (−0.459 + 0.796i)11-s − 2.17·12-s + (0.350 − 0.936i)13-s + 2.56·14-s + (0.886 − 1.53i)15-s + (−0.303 + 0.524i)16-s + (0.390 + 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.994 + 0.102i$
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: \((2,\ 403,\ (\ :1/2),\ 0.994 + 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16880 - 0.0598841i\)
\(L(\frac12)\) \(\approx\) \(1.16880 - 0.0598841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.26 + 3.37i)T \)
31 \( 1 - T \)
good2 \( 1 + (1.19 - 2.07i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.01 + 1.75i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.38T + 5T^{2} \)
7 \( 1 + (2.00 + 3.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.52 - 2.64i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.60 - 2.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.11 + 7.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.889 + 1.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.49 + 4.32i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (-1.86 + 3.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.73 - 9.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.47 - 6.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.521T + 47T^{2} \)
53 \( 1 + 3.40T + 53T^{2} \)
59 \( 1 + (-4.56 - 7.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.49 - 4.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.956 + 1.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.27 + 5.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.48T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + (0.951 - 1.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.26 - 14.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53472837605995191266494931904, −10.10404197637137379690758943406, −9.239897104625682772866176317990, −8.216512607528116880577151090985, −7.47220965859384125839357543992, −6.63737941398034407608893232662, −6.18354694694531255256737363122, −4.81960658245302443365292968231, −2.59912893148055988715152811818, −1.00340233022722042837470140397, 1.87805378873014131802321411380, 2.82069994548035387853102485256, 3.69032863553314233257228482329, 5.38118363440695440187030803685, 6.34858918726581488999083093827, 8.572614811507894385289750608066, 8.855722125031449331945489767013, 9.711018494267136532136746373994, 10.04194613489427184637182480720, 10.89703130596212303934986476285

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