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Properties

Label 2-430-43.11-c1-0-3
Degree $2$
Conductor $430$
Sign $0.791 - 0.611i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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  + (−0.623 − 0.781i)2-s + (−2.05 + 2.58i)3-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + 3.30·6-s + 3.39·7-s + (0.900 − 0.433i)8-s + (−1.75 − 7.70i)9-s + (−0.900 − 0.433i)10-s + (−0.325 − 1.42i)11-s + (−2.05 − 2.58i)12-s + (2.60 − 1.25i)13-s + (−2.11 − 2.65i)14-s + (−0.734 + 3.21i)15-s + (−0.900 − 0.433i)16-s + (5.38 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.440 − 0.552i)2-s + (−1.18 + 1.49i)3-s + (−0.111 + 0.487i)4-s + (0.402 − 0.194i)5-s + 1.34·6-s + 1.28·7-s + (0.318 − 0.153i)8-s + (−0.586 − 2.56i)9-s + (−0.284 − 0.137i)10-s + (−0.0982 − 0.430i)11-s + (−0.594 − 0.745i)12-s + (0.723 − 0.348i)13-s + (−0.565 − 0.708i)14-s + (−0.189 + 0.831i)15-s + (−0.225 − 0.108i)16-s + (1.30 + 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.791 - 0.611i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.791 - 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.877496 + 0.299686i\)
\(L(\frac12)\) \(\approx\) \(0.877496 + 0.299686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-6.13 + 2.31i)T \)
good3 \( 1 + (2.05 - 2.58i)T + (-0.667 - 2.92i)T^{2} \)
7 \( 1 - 3.39T + 7T^{2} \)
11 \( 1 + (0.325 + 1.42i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-2.60 + 1.25i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-5.38 - 2.59i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + (0.959 - 4.20i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-0.0151 - 0.0664i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (5.07 + 6.36i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-6.43 - 8.06i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + 5.45T + 37T^{2} \)
41 \( 1 + (-3.08 - 3.86i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (0.549 - 2.40i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-5.58 - 2.68i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (10.2 + 4.95i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (0.945 - 1.18i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (3.19 - 13.9i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-1.90 + 8.35i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-0.257 + 0.124i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + (-0.747 + 0.936i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (2.58 - 3.23i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-1.74 - 7.66i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.96282386424088907435033028684, −10.50747041440252002259024346904, −9.807084594083892014044753095275, −8.759814503089149724104546016893, −7.951949399212349920616332265557, −6.08071513226392729013576405473, −5.43909579991157757949875277365, −4.41652306905791470798660590491, −3.44720600150852673232703847847, −1.22824842619401177642514450173, 1.04471545062428391756459333579, 2.11254240226436241334362203469, 4.84891411620376327021720788846, 5.57002160969730831475508667063, 6.46350030146016050930173855586, 7.39480406071346914037891824009, 7.86553652951053190725302314637, 9.032440669435820396150939250657, 10.44728223687645757342724927701, 11.17534356761128159855782626771

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