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Properties

Label 2-429-13.4-c1-0-13
Degree $2$
Conductor $429$
Sign $0.162 - 0.986i$
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  + (2.17 + 1.25i)2-s + (0.5 − 0.866i)3-s + (2.16 + 3.74i)4-s + 2.19i·5-s + (2.17 − 1.25i)6-s + (−0.966 + 0.557i)7-s + 5.84i·8-s + (−0.499 − 0.866i)9-s + (−2.76 + 4.78i)10-s + (−0.866 − 0.5i)11-s + 4.32·12-s + (0.632 − 3.54i)13-s − 2.80·14-s + (1.90 + 1.09i)15-s + (−3.02 + 5.24i)16-s + (−0.814 − 1.41i)17-s + ⋯
L(s)  = 1  + (1.54 + 0.889i)2-s + (0.288 − 0.499i)3-s + (1.08 + 1.87i)4-s + 0.982i·5-s + (0.889 − 0.513i)6-s + (−0.365 + 0.210i)7-s + 2.06i·8-s + (−0.166 − 0.288i)9-s + (−0.873 + 1.51i)10-s + (−0.261 − 0.150i)11-s + 1.24·12-s + (0.175 − 0.984i)13-s − 0.749·14-s + (0.491 + 0.283i)15-s + (−0.756 + 1.31i)16-s + (−0.197 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.162 - 0.986i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.162 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49257 + 2.11505i\)
\(L(\frac12)\) \(\approx\) \(2.49257 + 2.11505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.632 + 3.54i)T \)
good2 \( 1 + (-2.17 - 1.25i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 2.19iT - 5T^{2} \)
7 \( 1 + (0.966 - 0.557i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (0.814 + 1.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.33 + 0.769i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.716 - 1.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.52 + 2.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.05iT - 31T^{2} \)
37 \( 1 + (-2.85 - 1.64i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.07 - 1.77i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.848 - 1.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.92iT - 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + (4.58 - 2.64i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.21 + 5.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.63 + 3.82i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.858 + 0.495i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.4iT - 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 + (3.97 + 2.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-16.1 + 9.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.65710317652681049511513852981, −10.81043008019151141222090785171, −9.481531294136458252407188991310, −7.967011811800004870536578890067, −7.49206832735958506407610626587, −6.39489333409998172796956632944, −5.94277860544258603660472069967, −4.66777586233668984196779953442, −3.26882453744150694529191583648, −2.73287450855158482797889562049, 1.62887418034890891791357604787, 3.04025322112744829831643121227, 4.13850971796944721032856884618, 4.76414818720165707468357864595, 5.72631753634426091493331049033, 6.87054752763721254759612354405, 8.464720229114852467398119635557, 9.413608169551097032748159057840, 10.32816558259516975508824905560, 11.13794588136424531861920230167

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