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Properties

Label 2-430-5.4-c1-0-9
Degree $2$
Conductor $430$
Sign $0.139 - 0.990i$
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  + i·2-s + 0.688i·3-s − 4-s + (0.311 − 2.21i)5-s − 0.688·6-s + 1.90i·7-s − i·8-s + 2.52·9-s + (2.21 + 0.311i)10-s + 0.214·11-s − 0.688i·12-s + 5.42i·13-s − 1.90·14-s + (1.52 + 0.214i)15-s + 16-s + 0.0666i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.397i·3-s − 0.5·4-s + (0.139 − 0.990i)5-s − 0.281·6-s + 0.719i·7-s − 0.353i·8-s + 0.841·9-s + (0.700 + 0.0983i)10-s + 0.0646·11-s − 0.198i·12-s + 1.50i·13-s − 0.508·14-s + (0.393 + 0.0553i)15-s + 0.250·16-s + 0.0161i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.139 - 0.990i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08147 + 0.940152i\)
\(L(\frac12)\) \(\approx\) \(1.08147 + 0.940152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-0.311 + 2.21i)T \)
43 \( 1 - iT \)
good3 \( 1 - 0.688iT - 3T^{2} \)
7 \( 1 - 1.90iT - 7T^{2} \)
11 \( 1 - 0.214T + 11T^{2} \)
13 \( 1 - 5.42iT - 13T^{2} \)
17 \( 1 - 0.0666iT - 17T^{2} \)
19 \( 1 - 7.90T + 19T^{2} \)
23 \( 1 - 4.21iT - 23T^{2} \)
29 \( 1 + 0.592T + 29T^{2} \)
31 \( 1 - 3.64T + 31T^{2} \)
37 \( 1 + 2.02iT - 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
47 \( 1 + 7.59iT - 47T^{2} \)
53 \( 1 + 8.57iT - 53T^{2} \)
59 \( 1 - 2.23T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 15.6iT - 67T^{2} \)
71 \( 1 - 6.16T + 71T^{2} \)
73 \( 1 + 15.2iT - 73T^{2} \)
79 \( 1 - 2.93T + 79T^{2} \)
83 \( 1 + 8.18iT - 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 2.06iT - 97T^{2} \)
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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

−11.69360596506177715200110760950, −10.03729698517738611740765831833, −9.378283624782373867974054255807, −8.823478181581336505491215404162, −7.67313692331953034390210433574, −6.71143853086647310023704717341, −5.47601049507832333045973775667, −4.80612364862519583032037581322, −3.72166524912093386471444064103, −1.60067863146845453800630682691, 1.09823879401322664863352925075, 2.74723234378950691606945550297, 3.67828743465090292793877691644, 5.04190850683663902689001361114, 6.34938846655497012609775697496, 7.38046819719718917385931104985, 7.979078382148097568778915189728, 9.543277167040835077113921362713, 10.28015200617835147944417564508, 10.74259364355696336607273650521

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