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Properties

Label 2-429-33.32-c1-0-33
Degree $2$
Conductor $429$
Sign $0.999 + 0.0431i$
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  + 2.70·2-s + (−1.68 − 0.380i)3-s + 5.32·4-s + 2.40i·5-s + (−4.57 − 1.02i)6-s − 4.64i·7-s + 8.98·8-s + (2.71 + 1.28i)9-s + 6.49i·10-s + (0.588 + 3.26i)11-s + (−8.98 − 2.02i)12-s − i·13-s − 12.5i·14-s + (0.913 − 4.05i)15-s + 13.6·16-s − 2.96·17-s + ⋯
L(s)  = 1  + 1.91·2-s + (−0.975 − 0.219i)3-s + 2.66·4-s + 1.07i·5-s + (−1.86 − 0.420i)6-s − 1.75i·7-s + 3.17·8-s + (0.903 + 0.428i)9-s + 2.05i·10-s + (0.177 + 0.984i)11-s + (−2.59 − 0.584i)12-s − 0.277i·13-s − 3.36i·14-s + (0.235 − 1.04i)15-s + 3.41·16-s − 0.720·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.999 + 0.0431i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.999 + 0.0431i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.25824 - 0.0703876i\)
\(L(\frac12)\) \(\approx\) \(3.25824 - 0.0703876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 + 0.380i)T \)
11 \( 1 + (-0.588 - 3.26i)T \)
13 \( 1 + iT \)
good2 \( 1 - 2.70T + 2T^{2} \)
5 \( 1 - 2.40iT - 5T^{2} \)
7 \( 1 + 4.64iT - 7T^{2} \)
17 \( 1 + 2.96T + 17T^{2} \)
19 \( 1 - 0.907iT - 19T^{2} \)
23 \( 1 - 0.745iT - 23T^{2} \)
29 \( 1 + 4.15T + 29T^{2} \)
31 \( 1 + 2.63T + 31T^{2} \)
37 \( 1 + 9.06T + 37T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 + 6.71iT - 43T^{2} \)
47 \( 1 + 6.35iT - 47T^{2} \)
53 \( 1 - 6.81iT - 53T^{2} \)
59 \( 1 - 5.61iT - 59T^{2} \)
61 \( 1 - 5.76iT - 61T^{2} \)
67 \( 1 - 8.08T + 67T^{2} \)
71 \( 1 + 0.0134iT - 71T^{2} \)
73 \( 1 - 4.80iT - 73T^{2} \)
79 \( 1 + 8.17iT - 79T^{2} \)
83 \( 1 - 1.87T + 83T^{2} \)
89 \( 1 + 6.88iT - 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−11.31652208945205819331290601337, −10.53099178203499542889350054417, −10.28344617334706914355578942808, −7.34414007952807865014298603268, −7.15535987339475718073471569043, −6.44648356518682510176941690307, −5.26596071706941939072691037383, −4.29477946742309262145719312185, −3.54748793598926421976525901414, −1.88428880133822898888122908331, 1.90739511631024023094209599564, 3.43524143312232541129287519211, 4.71671575263020707837031819193, 5.28750837087636141999166310553, 5.96689764739987540485413306733, 6.74981115644671616888257000204, 8.399812438415503560631047680466, 9.408146255639161347975866891931, 11.01169102444468204799725383911, 11.47077181272280738875056666438

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