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Properties

Label 2-430-43.16-c1-0-4
Degree $2$
Conductor $430$
Sign $0.999 - 0.0358i$
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  + (−0.222 − 0.974i)2-s + (−0.171 + 0.752i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + 0.771·6-s − 3.65·7-s + (0.623 + 0.781i)8-s + (2.16 + 1.04i)9-s + (−0.623 + 0.781i)10-s + (5.55 + 2.67i)11-s + (−0.171 − 0.752i)12-s + (0.656 + 0.823i)13-s + (0.812 + 3.55i)14-s + (0.695 − 0.334i)15-s + (0.623 − 0.781i)16-s + (1.60 − 2.01i)17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (−0.0991 + 0.434i)3-s + (−0.450 + 0.216i)4-s + (−0.278 − 0.349i)5-s + 0.315·6-s − 1.38·7-s + (0.220 + 0.276i)8-s + (0.722 + 0.347i)9-s + (−0.197 + 0.247i)10-s + (1.67 + 0.806i)11-s + (−0.0495 − 0.217i)12-s + (0.182 + 0.228i)13-s + (0.217 + 0.951i)14-s + (0.179 − 0.0864i)15-s + (0.155 − 0.195i)16-s + (0.389 − 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.999 - 0.0358i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.999 - 0.0358i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11486 + 0.0199938i\)
\(L(\frac12)\) \(\approx\) \(1.11486 + 0.0199938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-6.55 + 0.0169i)T \)
good3 \( 1 + (0.171 - 0.752i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 + 3.65T + 7T^{2} \)
11 \( 1 + (-5.55 - 2.67i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-0.656 - 0.823i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.60 + 2.01i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + (-2.29 + 1.10i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-4.64 - 2.23i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-0.623 - 2.73i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-1.65 - 7.24i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 - 0.188T + 37T^{2} \)
41 \( 1 + (1.94 + 8.52i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (3.66 - 1.76i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (3.38 - 4.24i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (4.36 - 5.47i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-2.49 + 10.9i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-5.97 + 2.87i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (7.41 - 3.57i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-0.110 - 0.138i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 1.72T + 79T^{2} \)
83 \( 1 + (-0.708 + 3.10i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-1.21 + 5.31i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (3.65 + 1.76i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.13118216911619574349237052079, −10.13638608276756624446154460780, −9.374819686665307768689459569823, −9.032640241618433309806481551939, −7.35172364348663906092209741333, −6.65253522764538271070534334016, −5.08974391326381677216410285311, −4.07696370999818615436272888915, −3.22265397665175220235184932460, −1.34341575130341926869208520566, 0.949482736151038909453373545895, 3.28280948726318183845441897088, 4.14600527844738065121475102693, 6.01872194102654092268327592431, 6.44177337887058131862681009424, 7.21666742635080041462207742708, 8.323470202144274028307678795595, 9.418748610199292127694862151716, 9.904836146077783802917028385959, 11.21189014638011289475143073756

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