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Properties

Label 2-430-43.6-c1-0-15
Degree $2$
Conductor $430$
Sign $0.214 + 0.976i$
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  + 2-s + (1.25 − 2.17i)3-s + 4-s + (0.5 − 0.866i)5-s + (1.25 − 2.17i)6-s + (−0.660 − 1.14i)7-s + 8-s + (−1.66 − 2.87i)9-s + (0.5 − 0.866i)10-s − 3.51·11-s + (1.25 − 2.17i)12-s + (2.56 + 4.44i)13-s + (−0.660 − 1.14i)14-s + (−1.25 − 2.17i)15-s + 16-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.725 − 1.25i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.513 − 0.888i)6-s + (−0.249 − 0.432i)7-s + 0.353·8-s + (−0.553 − 0.958i)9-s + (0.158 − 0.273i)10-s − 1.05·11-s + (0.362 − 0.628i)12-s + (0.711 + 1.23i)13-s + (−0.176 − 0.305i)14-s + (−0.324 − 0.562i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.214 + 0.976i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.214 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99784 - 1.60591i\)
\(L(\frac12)\) \(\approx\) \(1.99784 - 1.60591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (5.94 - 2.76i)T \)
good3 \( 1 + (-1.25 + 2.17i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.660 + 1.14i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.51T + 11T^{2} \)
13 \( 1 + (-2.56 - 4.44i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.757 + 1.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.66 - 4.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.43 - 5.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.09 + 1.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.75 + 6.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.613T + 41T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + (-1.80 + 3.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.67T + 59T^{2} \)
61 \( 1 + (-2.85 - 4.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.38 - 9.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.41 + 7.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.73 - 4.74i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.61 + 2.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.78 + 3.10i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.04 - 15.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−11.26104489227374785702570587792, −10.08685589978125492384075166734, −8.926254847900459361829049842158, −8.028226895286605639115852968434, −7.17341014902537711746994137968, −6.41832557061804472206425741081, −5.23788733242893129642013738982, −3.89149469927929733131437079833, −2.61532056083640891476069828479, −1.47264135467152118990940454095, 2.62893135427114816730898397283, 3.26340961103272476518052453666, 4.44347058060883957495022609954, 5.44924750357311791524985423494, 6.35819331645536094534071904328, 7.932063655012882413043099879096, 8.551923501309564728265878877917, 9.965019269131970927011595862821, 10.23400219627615041439636163604, 11.15488963515792025604212576622

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