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Properties

Label 2-403-403.36-c1-0-8
Degree $2$
Conductor $403$
Sign $-0.0509 - 0.998i$
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  + (−0.605 + 0.349i)2-s − 2.90·3-s + (−0.755 + 1.30i)4-s + (1.79 − 1.03i)5-s + (1.75 − 1.01i)6-s + (−0.132 + 0.0763i)7-s − 2.45i·8-s + 5.41·9-s + (−0.725 + 1.25i)10-s + (−1.09 − 0.630i)11-s + (2.19 − 3.79i)12-s + (2.96 − 2.04i)13-s + (0.0534 − 0.0925i)14-s + (−5.20 + 3.00i)15-s + (−0.651 − 1.12i)16-s + (1.94 + 3.37i)17-s + ⋯
L(s)  = 1  + (−0.428 + 0.247i)2-s − 1.67·3-s + (−0.377 + 0.654i)4-s + (0.802 − 0.463i)5-s + (0.717 − 0.414i)6-s + (−0.0499 + 0.0288i)7-s − 0.868i·8-s + 1.80·9-s + (−0.229 + 0.397i)10-s + (−0.329 − 0.190i)11-s + (0.632 − 1.09i)12-s + (0.823 − 0.567i)13-s + (0.0142 − 0.0247i)14-s + (−1.34 + 0.776i)15-s + (−0.162 − 0.282i)16-s + (0.472 + 0.818i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0509 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0509 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.0509 - 0.998i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.0509 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364989 + 0.384073i\)
\(L(\frac12)\) \(\approx\) \(0.364989 + 0.384073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.96 + 2.04i)T \)
31 \( 1 + (-5.51 - 0.768i)T \)
good2 \( 1 + (0.605 - 0.349i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 2.90T + 3T^{2} \)
5 \( 1 + (-1.79 + 1.03i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.132 - 0.0763i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.09 + 0.630i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.94 - 3.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.86 - 3.96i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.11 - 7.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.981 + 1.69i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 - 9.54iT - 37T^{2} \)
41 \( 1 + (1.93 + 1.11i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.63 + 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.61iT - 47T^{2} \)
53 \( 1 + (-3.09 - 5.36i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.21 - 1.28i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.54 + 6.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.78 - 5.07i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + (-3.69 + 2.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.25 + 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.84 + 2.22i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.542 + 0.313i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.09 - 1.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.48101342704799399960136780073, −10.47229911602368401462823871927, −9.895349147788479492137085681385, −8.708212054003458249131453275078, −7.84263661733238750963149687414, −6.52252863629456546976652457110, −5.86288090119431950232684924333, −4.96360180132735638113066885275, −3.69065345126078592595150899061, −1.25163950705955596839337274721, 0.57507897156702957468670580378, 2.18050297473630173545153040835, 4.53715610112411466140174468744, 5.27441648013193064525052739234, 6.35069458180604027095345074475, 6.70286386739932105905612664967, 8.515305826919746154732284727580, 9.563718363610751064710573018768, 10.37766029150942210091028977890, 10.87370811550975081247797127351

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