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Properties

Label 2-430-43.16-c1-0-9
Degree $2$
Conductor $430$
Sign $0.954 + 0.298i$
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 + (0.565 − 2.47i)3-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + 2.54·6-s + 2.52·7-s + (−0.623 − 0.781i)8-s + (−3.12 − 1.50i)9-s + (−0.623 + 0.781i)10-s + (2.46 + 1.18i)11-s + (0.565 + 2.47i)12-s + (−0.821 − 1.03i)13-s + (0.562 + 2.46i)14-s + (2.29 − 1.10i)15-s + (0.623 − 0.781i)16-s + (3.17 − 3.98i)17-s + ⋯
L(s)  = 1  + (0.157 + 0.689i)2-s + (0.326 − 1.43i)3-s + (−0.450 + 0.216i)4-s + (0.278 + 0.349i)5-s + 1.03·6-s + 0.954·7-s + (−0.220 − 0.276i)8-s + (−1.04 − 0.501i)9-s + (−0.197 + 0.247i)10-s + (0.743 + 0.358i)11-s + (0.163 + 0.715i)12-s + (−0.227 − 0.285i)13-s + (0.150 + 0.658i)14-s + (0.591 − 0.284i)15-s + (0.155 − 0.195i)16-s + (0.770 − 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.954 + 0.298i$
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: \((2,\ 430,\ (\ :1/2),\ 0.954 + 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81319 - 0.277397i\)
\(L(\frac12)\) \(\approx\) \(1.81319 - 0.277397i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.18 - 2.16i)T \)
good3 \( 1 + (-0.565 + 2.47i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 - 2.52T + 7T^{2} \)
11 \( 1 + (-2.46 - 1.18i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.821 + 1.03i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-3.17 + 3.98i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + (2.36 - 1.13i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (1.44 + 0.693i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (0.451 + 1.97i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-0.223 - 0.981i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 - 4.40T + 37T^{2} \)
41 \( 1 + (0.447 + 1.96i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (9.74 - 4.69i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.437 + 0.548i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.26 - 1.59i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (3.03 - 13.2i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (11.4 - 5.50i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (6.84 - 3.29i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-4.21 - 5.29i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 5.12T + 79T^{2} \)
83 \( 1 + (1.67 - 7.34i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-0.521 + 2.28i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-11.5 - 5.57i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.45246744475290327480063298564, −10.08182787566349996004524016676, −8.984150661017943766739752243519, −7.968128455187167006594505444308, −7.46577424751375958154715946212, −6.60355912007046183270149289155, −5.69396171926053616002374161078, −4.40917508667479493872305086235, −2.71258613170031493439941211787, −1.37274748718395224867987858545, 1.71770381957008772166583652517, 3.35166389012293031684356117922, 4.29062340625924248422719732170, 4.98915287339808560643525255778, 6.12420392490382608762230807781, 7.972260536989041664493763585003, 8.822230186119748923623541347264, 9.499960754696087452908607762854, 10.33872319388589710192824047668, 11.05063894779897560252236071664

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