Properties

Label 2-430-5.4-c1-0-13
Degree $2$
Conductor $430$
Sign $0.0921 + 0.995i$
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 + 0.310i·3-s − 4-s + (−0.205 − 2.22i)5-s + 0.310·6-s + 2.43i·7-s + i·8-s + 2.90·9-s + (−2.22 + 0.205i)10-s + 4.13·11-s − 0.310i·12-s − 2.98i·13-s + 2.43·14-s + (0.692 − 0.0640i)15-s + 16-s − 8.05i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.179i·3-s − 0.5·4-s + (−0.0921 − 0.995i)5-s + 0.126·6-s + 0.921i·7-s + 0.353i·8-s + 0.967·9-s + (−0.704 + 0.0651i)10-s + 1.24·11-s − 0.0897i·12-s − 0.828i·13-s + 0.651·14-s + (0.178 − 0.0165i)15-s + 0.250·16-s − 1.95i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0921 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0921 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.0921 + 0.995i$
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: \((2,\ 430,\ (\ :1/2),\ 0.0921 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03601 - 0.944602i\)
\(L(\frac12)\) \(\approx\) \(1.03601 - 0.944602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.205 + 2.22i)T \)
43 \( 1 - iT \)
good3 \( 1 - 0.310iT - 3T^{2} \)
7 \( 1 - 2.43iT - 7T^{2} \)
11 \( 1 - 4.13T + 11T^{2} \)
13 \( 1 + 2.98iT - 13T^{2} \)
17 \( 1 + 8.05iT - 17T^{2} \)
19 \( 1 + 4.02T + 19T^{2} \)
23 \( 1 + 1.71iT - 23T^{2} \)
29 \( 1 + 1.98T + 29T^{2} \)
31 \( 1 - 6.12T + 31T^{2} \)
37 \( 1 + 1.06iT - 37T^{2} \)
41 \( 1 - 8.13T + 41T^{2} \)
47 \( 1 - 0.297iT - 47T^{2} \)
53 \( 1 - 13.0iT - 53T^{2} \)
59 \( 1 + 0.202T + 59T^{2} \)
61 \( 1 + 3.83T + 61T^{2} \)
67 \( 1 - 6.74iT - 67T^{2} \)
71 \( 1 - 5.40T + 71T^{2} \)
73 \( 1 + 1.14iT - 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 - 3.57iT - 83T^{2} \)
89 \( 1 + 3.88T + 89T^{2} \)
97 \( 1 - 4.77iT - 97T^{2} \)
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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

−11.07937318726377937727557184310, −9.859606733657058922465583064903, −9.256377131476391214971182122179, −8.589866313354156867829869604635, −7.38024053404602570744071039752, −5.98690086201409945856060895452, −4.86382829910439403941797599781, −4.11557897435769398780901085450, −2.59018087549030462927488915660, −1.05397673305660642104890810193, 1.65829932777762232156198728972, 3.86802641468756696216927753021, 4.22260657510173830438164067581, 6.22647284857184157439431026540, 6.65454228796806420072076621708, 7.44940366173416502005146239655, 8.436901756960841720020301960005, 9.645605367423868361090685734820, 10.38630491915453894691067667125, 11.22069553457714652392847263053

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