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Properties

Label 2-403-403.398-c1-0-30
Degree $2$
Conductor $403$
Sign $0.994 + 0.106i$
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  + (0.366 + 0.366i)2-s + (2.36 + 1.36i)3-s − 1.73i·4-s + (0.366 − 1.36i)5-s + (0.366 + 1.36i)6-s + (−0.767 − 2.86i)7-s + (1.36 − 1.36i)8-s + (2.23 + 3.86i)9-s + (0.633 − 0.366i)10-s + (−0.232 + 0.866i)11-s + (2.36 − 4.09i)12-s + (−1.59 + 3.23i)13-s + (0.767 − 1.33i)14-s + (2.73 − 2.73i)15-s − 2.46·16-s + ⋯
L(s)  = 1  + (0.258 + 0.258i)2-s + (1.36 + 0.788i)3-s − 0.866i·4-s + (0.163 − 0.610i)5-s + (0.149 + 0.557i)6-s + (−0.290 − 1.08i)7-s + (0.482 − 0.482i)8-s + (0.744 + 1.28i)9-s + (0.200 − 0.115i)10-s + (−0.0699 + 0.261i)11-s + (0.683 − 1.18i)12-s + (−0.443 + 0.896i)13-s + (0.205 − 0.355i)14-s + (0.705 − 0.705i)15-s − 0.616·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.994 + 0.106i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.994 + 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35356 - 0.126181i\)
\(L(\frac12)\) \(\approx\) \(2.35356 - 0.126181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.59 - 3.23i)T \)
31 \( 1 + (-4.33 - 3.5i)T \)
good2 \( 1 + (-0.366 - 0.366i)T + 2iT^{2} \)
3 \( 1 + (-2.36 - 1.36i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.366 + 1.36i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.767 + 2.86i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.232 - 0.866i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.96 + 1.33i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 6.19T + 23T^{2} \)
29 \( 1 - 8.46iT - 29T^{2} \)
37 \( 1 + (4.73 - 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.53 - 5.73i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.09 - 3.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.53 + 1.53i)T - 47iT^{2} \)
53 \( 1 + (5.76 - 3.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.562 - 2.09i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 3.19iT - 61T^{2} \)
67 \( 1 + (-2.86 + 10.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.330 - 1.23i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.83 + 10.5i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (10.7 + 6.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.5 - 4.16i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.26 + 2.26i)T + 89iT^{2} \)
97 \( 1 + (8.46 + 8.46i)T + 97iT^{2} \)
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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

−10.87890575868687229319068663623, −9.980528163055860181150204236645, −9.571213415306689233290269111679, −8.724495125058795399482674318588, −7.53098116406217798335880917742, −6.64418530187372659717682553437, −5.03275369666209040857112517831, −4.42577085590612606276828839841, −3.27827021633133061898034685566, −1.57253453361196705947115387063, 2.30800384652830737695782049785, 2.79772557084370523612026589067, 3.77324730109972719353251727106, 5.57967274426101864254325791023, 6.82416774919081438415016428667, 7.85954894024810863011227386949, 8.246446387793109322512084710025, 9.273195209283438413896738133495, 10.23046119959422155673723921029, 11.72087398604836664656977932649

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