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Properties

Label 2-430-43.13-c1-0-2
Degree $2$
Conductor $430$
Sign $0.451 - 0.892i$
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 + (−0.419 + 0.389i)3-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (0.286 + 0.495i)6-s + (−0.761 + 1.31i)7-s + (−0.623 + 0.781i)8-s + (−0.199 + 2.66i)9-s + (−0.365 + 0.930i)10-s + (−0.360 + 0.173i)11-s + (0.547 − 0.168i)12-s + (0.927 + 2.36i)13-s + (1.11 + 1.03i)14-s + (0.473 − 0.322i)15-s + (0.623 + 0.781i)16-s + (−1.75 + 0.264i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (−0.242 + 0.224i)3-s + (−0.450 − 0.216i)4-s + (−0.442 − 0.0666i)5-s + (0.116 + 0.202i)6-s + (−0.287 + 0.498i)7-s + (−0.220 + 0.276i)8-s + (−0.0665 + 0.888i)9-s + (−0.115 + 0.294i)10-s + (−0.108 + 0.0523i)11-s + (0.157 − 0.0487i)12-s + (0.257 + 0.655i)13-s + (0.298 + 0.276i)14-s + (0.122 − 0.0832i)15-s + (0.155 + 0.195i)16-s + (−0.425 + 0.0641i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.451 - 0.892i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742203 + 0.456083i\)
\(L(\frac12)\) \(\approx\) \(0.742203 + 0.456083i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.988 + 0.149i)T \)
43 \( 1 + (5.37 - 3.75i)T \)
good3 \( 1 + (0.419 - 0.389i)T + (0.224 - 2.99i)T^{2} \)
7 \( 1 + (0.761 - 1.31i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.360 - 0.173i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.927 - 2.36i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (1.75 - 0.264i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.403 - 5.38i)T + (-18.7 + 2.83i)T^{2} \)
23 \( 1 + (-1.19 - 0.813i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (-3.23 - 3.00i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (1.25 - 0.388i)T + (25.6 - 17.4i)T^{2} \)
37 \( 1 + (0.123 + 0.213i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.104 - 0.459i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (7.59 + 3.65i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (0.426 - 1.08i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (1.61 + 2.02i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-3.96 - 1.22i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (0.696 + 9.29i)T + (-66.2 + 9.98i)T^{2} \)
71 \( 1 + (-6.48 + 4.42i)T + (25.9 - 66.0i)T^{2} \)
73 \( 1 + (0.654 + 1.66i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (1.08 - 1.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.814 + 0.755i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (-1.54 + 1.43i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (-0.384 + 0.185i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.32193735645987655367227544278, −10.57510877502984415220781083933, −9.692348927234743406865258910626, −8.697046515797883101162154201225, −7.84721316721558443883656207940, −6.48644680349795937920648561402, −5.35282518981502280175232509481, −4.43540501191407375125859427877, −3.26461122036095882104298351000, −1.86006106790673733345116587004, 0.54800735055082127747642164938, 3.08392835439729725307535584524, 4.17421930806542911131179349986, 5.34562413144014776020259760989, 6.52752997794595361691182101342, 7.04458833914466667417070195011, 8.146060565235573276558067154753, 9.022421309588250139462027534035, 10.03185111298044623003053829989, 11.11190245939699525125309338429

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