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Properties

Label 2-403-13.3-c1-0-9
Degree $2$
Conductor $403$
Sign $-0.185 - 0.982i$
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  + (1.20 + 2.09i)2-s + (−0.882 − 1.52i)3-s + (−1.91 + 3.31i)4-s + 3.22·5-s + (2.12 − 3.68i)6-s + (−1.47 + 2.55i)7-s − 4.40·8-s + (−0.0574 + 0.0995i)9-s + (3.89 + 6.74i)10-s + (0.649 + 1.12i)11-s + 6.74·12-s + (3.25 − 1.54i)13-s − 7.11·14-s + (−2.84 − 4.92i)15-s + (−1.48 − 2.57i)16-s + (−1.01 + 1.76i)17-s + ⋯
L(s)  = 1  + (0.853 + 1.47i)2-s + (−0.509 − 0.882i)3-s + (−0.956 + 1.65i)4-s + 1.44·5-s + (0.869 − 1.50i)6-s + (−0.556 + 0.964i)7-s − 1.55·8-s + (−0.0191 + 0.0331i)9-s + (1.23 + 2.13i)10-s + (0.195 + 0.338i)11-s + 1.94·12-s + (0.903 − 0.427i)13-s − 1.90·14-s + (−0.734 − 1.27i)15-s + (−0.372 − 0.644i)16-s + (−0.246 + 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.185 - 0.982i$
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: \((2,\ 403,\ (\ :1/2),\ -0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33614 + 1.61167i\)
\(L(\frac12)\) \(\approx\) \(1.33614 + 1.61167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.25 + 1.54i)T \)
31 \( 1 - T \)
good2 \( 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.882 + 1.52i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.22T + 5T^{2} \)
7 \( 1 + (1.47 - 2.55i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.649 - 1.12i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.01 - 1.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.45 - 2.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.65 + 2.86i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (4.16 + 7.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.79 + 6.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.82 + 10.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.30T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (2.23 - 3.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.15 + 7.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.67 + 8.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.66 + 4.61i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 8.52T + 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 + (-1.52 - 2.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.01 - 15.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.17853839794868671539772637809, −10.59193055280788917946789097283, −9.349261856208877391046587773160, −8.585838802909725623040053236868, −7.29345537121670865127790699229, −6.49888378559894419345486411681, −5.77470511262251276372345186999, −5.56171333102580183839581358468, −3.77790647307278947666087008340, −1.97185609053701287614345321120, 1.34586456963396396530681872356, 2.82014251386660870932866432205, 4.01125865962980541949393168607, 4.82026057395438255208874804682, 5.79420351360947800209235564235, 6.75680381406320138487329430180, 8.922803602867335808690582998166, 9.813236561860444725893008653759, 10.27822802733060561143451086549, 10.94584336550038777079922531966

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