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Properties

Label 2-430-215.128-c1-0-16
Degree $2$
Conductor $430$
Sign $0.999 + 0.0378i$
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  + (0.707 + 0.707i)2-s + (1.12 − 1.12i)3-s + 1.00i·4-s + (1.89 − 1.19i)5-s + 1.59·6-s + (−0.600 − 0.600i)7-s + (−0.707 + 0.707i)8-s + 0.453i·9-s + (2.18 + 0.491i)10-s + 3.86·11-s + (1.12 + 1.12i)12-s + (−3.06 − 3.06i)13-s − 0.848i·14-s + (0.784 − 3.48i)15-s − 1.00·16-s + (−3.44 + 3.44i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.651 − 0.651i)3-s + 0.500i·4-s + (0.845 − 0.534i)5-s + 0.651·6-s + (−0.226 − 0.226i)7-s + (−0.250 + 0.250i)8-s + 0.151i·9-s + (0.689 + 0.155i)10-s + 1.16·11-s + (0.325 + 0.325i)12-s + (−0.850 − 0.850i)13-s − 0.226i·14-s + (0.202 − 0.898i)15-s − 0.250·16-s + (−0.834 + 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.999 + 0.0378i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.999 + 0.0378i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42073 - 0.0458255i\)
\(L(\frac12)\) \(\approx\) \(2.42073 - 0.0458255i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.89 + 1.19i)T \)
43 \( 1 + (4.85 - 4.40i)T \)
good3 \( 1 + (-1.12 + 1.12i)T - 3iT^{2} \)
7 \( 1 + (0.600 + 0.600i)T + 7iT^{2} \)
11 \( 1 - 3.86T + 11T^{2} \)
13 \( 1 + (3.06 + 3.06i)T + 13iT^{2} \)
17 \( 1 + (3.44 - 3.44i)T - 17iT^{2} \)
19 \( 1 - 5.26T + 19T^{2} \)
23 \( 1 + (2.29 + 2.29i)T + 23iT^{2} \)
29 \( 1 + 1.15T + 29T^{2} \)
31 \( 1 + 8.29T + 31T^{2} \)
37 \( 1 + (-2.98 - 2.98i)T + 37iT^{2} \)
41 \( 1 - 6.25T + 41T^{2} \)
47 \( 1 + (8.96 - 8.96i)T - 47iT^{2} \)
53 \( 1 + (0.712 + 0.712i)T + 53iT^{2} \)
59 \( 1 - 6.93iT - 59T^{2} \)
61 \( 1 + 0.0822iT - 61T^{2} \)
67 \( 1 + (4.42 - 4.42i)T - 67iT^{2} \)
71 \( 1 + 6.42iT - 71T^{2} \)
73 \( 1 + (0.675 - 0.675i)T - 73iT^{2} \)
79 \( 1 - 8.61iT - 79T^{2} \)
83 \( 1 + (6.64 + 6.64i)T + 83iT^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (0.895 - 0.895i)T - 97iT^{2} \)
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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

−11.31817963879262410872933252346, −10.02984696747304621235069243146, −9.181291216660572860100504159346, −8.291550286312994274851523409382, −7.40736363881158996638730723803, −6.48543391067056241461398683500, −5.51674837941524670886522645483, −4.39796497621989798849587692101, −2.96695366866227114857270522438, −1.65543104912272816919179146882, 1.95361947883743196488819475949, 3.10039796913662603953314378302, 4.04535845828903287290390051434, 5.23936148235380408144864244634, 6.40444693834063338028387247328, 7.22491839243567278197876948522, 9.196856902543233398189045934335, 9.292002936266521724557154643664, 10.04021208587307856449394258398, 11.30611630671604475512457656113

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