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Properties

Label 2-429-143.18-c1-0-15
Degree $2$
Conductor $429$
Sign $0.962 - 0.271i$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
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  + (0.123 + 0.778i)2-s + (−0.309 + 0.951i)3-s + (1.31 − 0.426i)4-s + (0.499 − 3.15i)5-s + (−0.778 − 0.123i)6-s + (−0.533 − 0.271i)7-s + (1.20 + 2.37i)8-s + (−0.809 − 0.587i)9-s + 2.51·10-s + (−0.840 + 3.20i)11-s + 1.37i·12-s + (2.14 − 2.90i)13-s + (0.145 − 0.448i)14-s + (2.84 + 1.44i)15-s + (0.533 − 0.387i)16-s + (3.69 − 2.68i)17-s + ⋯
L(s)  = 1  + (0.0871 + 0.550i)2-s + (−0.178 + 0.549i)3-s + (0.655 − 0.213i)4-s + (0.223 − 1.41i)5-s + (−0.317 − 0.0503i)6-s + (−0.201 − 0.102i)7-s + (0.427 + 0.838i)8-s + (−0.269 − 0.195i)9-s + 0.795·10-s + (−0.253 + 0.967i)11-s + 0.398i·12-s + (0.594 − 0.804i)13-s + (0.0389 − 0.119i)14-s + (0.734 + 0.374i)15-s + (0.133 − 0.0969i)16-s + (0.895 − 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70817 + 0.236027i\)
\(L(\frac12)\) \(\approx\) \(1.70817 + 0.236027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.840 - 3.20i)T \)
13 \( 1 + (-2.14 + 2.90i)T \)
good2 \( 1 + (-0.123 - 0.778i)T + (-1.90 + 0.618i)T^{2} \)
5 \( 1 + (-0.499 + 3.15i)T + (-4.75 - 1.54i)T^{2} \)
7 \( 1 + (0.533 + 0.271i)T + (4.11 + 5.66i)T^{2} \)
17 \( 1 + (-3.69 + 2.68i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.84 + 2.46i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 - 1.52iT - 23T^{2} \)
29 \( 1 + (3.77 - 1.22i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.82 + 0.763i)T + (29.4 - 9.57i)T^{2} \)
37 \( 1 + (4.48 - 8.80i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (4.44 - 2.26i)T + (24.0 - 33.1i)T^{2} \)
43 \( 1 - 4.12T + 43T^{2} \)
47 \( 1 + (10.4 - 5.34i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (8.15 + 5.92i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.743 - 1.45i)T + (-34.6 - 47.7i)T^{2} \)
61 \( 1 + (3.98 + 5.48i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.22 - 4.22i)T - 67iT^{2} \)
71 \( 1 + (-0.0218 + 0.137i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 + (-12.4 - 6.34i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-3.28 + 4.52i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.68 + 1.05i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (3.55 + 3.55i)T + 89iT^{2} \)
97 \( 1 + (9.89 - 1.56i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.28495872424803453428974713189, −10.04361972411134424603113598794, −9.578891895898308333919629785698, −8.342163818004666754895661688475, −7.56679595787330784018074767673, −6.37334325301588509390719624415, −5.15451327887762486338281827003, −4.99447454786246359442046167321, −3.17353899194612981598469002051, −1.33044415328583032712840127966, 1.65971113046234031598628295316, 2.95817970480854421881624483644, 3.62246720455498620429882349646, 5.80010915251864625770848057248, 6.43138846220887673601301332862, 7.28851852385841858143004377931, 8.147471689250599170022949040767, 9.631091553031652001286180934089, 10.61076253148108394460284996826, 11.07896943847599393990864811526

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