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Properties

Label 2-3-3.2-c70-0-19
Degree $2$
Conductor $3$
Sign $-0.981 - 0.191i$
Analytic cond. $93.0951$
Root an. cond. $9.64858$
Motivic weight $70$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.57e10i·2-s + (4.91e16 + 9.56e15i)3-s − 9.10e20·4-s − 1.29e24i·5-s + (4.37e26 − 2.24e27i)6-s + 1.42e29·7-s − 1.23e31i·8-s + (2.32e33 + 9.39e32i)9-s − 5.92e34·10-s + 9.16e35i·11-s + (−4.47e37 − 8.71e36i)12-s − 1.00e38·13-s − 6.52e39i·14-s + (1.23e40 − 6.36e40i)15-s − 1.63e42·16-s − 1.11e43i·17-s + ⋯
L(s)  = 1  − 1.33i·2-s + (0.981 + 0.191i)3-s − 0.771·4-s − 0.445i·5-s + (0.254 − 1.30i)6-s + 0.376·7-s − 0.304i·8-s + (0.926 + 0.375i)9-s − 0.592·10-s + 0.326i·11-s + (−0.756 − 0.147i)12-s − 0.103·13-s − 0.501i·14-s + (0.0851 − 0.436i)15-s − 1.17·16-s − 0.958i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.981 - 0.191i$
Analytic conductor: \(93.0951\)
Root analytic conductor: \(9.64858\)
Motivic weight: \(70\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :35),\ -0.981 - 0.191i)\)

Particular Values

\(L(\frac{71}{2})\) \(\approx\) \(2.800563617\)
\(L(\frac12)\) \(\approx\) \(2.800563617\)
\(L(36)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.91e16 - 9.56e15i)T \)
good2 \( 1 + 4.57e10iT - 1.18e21T^{2} \)
5 \( 1 + 1.29e24iT - 8.47e48T^{2} \)
7 \( 1 - 1.42e29T + 1.43e59T^{2} \)
11 \( 1 - 9.16e35iT - 7.89e72T^{2} \)
13 \( 1 + 1.00e38T + 9.46e77T^{2} \)
17 \( 1 + 1.11e43iT - 1.35e86T^{2} \)
19 \( 1 + 3.22e44T + 3.25e89T^{2} \)
23 \( 1 + 1.73e47iT - 2.09e95T^{2} \)
29 \( 1 + 8.83e50iT - 2.33e102T^{2} \)
31 \( 1 + 2.36e52T + 2.48e104T^{2} \)
37 \( 1 - 3.62e54T + 5.94e109T^{2} \)
41 \( 1 + 2.87e56iT - 7.85e112T^{2} \)
43 \( 1 - 1.69e57T + 2.20e114T^{2} \)
47 \( 1 + 4.55e58iT - 1.11e117T^{2} \)
53 \( 1 + 3.77e60iT - 5.00e120T^{2} \)
59 \( 1 - 4.10e61iT - 9.11e123T^{2} \)
61 \( 1 - 1.22e62T + 9.39e124T^{2} \)
67 \( 1 + 4.53e63T + 6.68e127T^{2} \)
71 \( 1 + 1.05e65iT - 3.87e129T^{2} \)
73 \( 1 + 2.52e65T + 2.70e130T^{2} \)
79 \( 1 + 1.05e65T + 6.82e132T^{2} \)
83 \( 1 - 2.76e67iT - 2.16e134T^{2} \)
89 \( 1 + 6.66e67iT - 2.86e136T^{2} \)
97 \( 1 - 5.39e68T + 1.18e139T^{2} \)
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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

−12.33346215924918059404255287496, −10.90942264519259938821912767163, −9.700631133896861341069910956123, −8.711859405236168856417335863919, −7.17897324095654021928287688658, −4.82622819410393706620642711012, −3.78458990646883382853407831889, −2.55800832222278062331706820184, −1.73839073865209700073966998883, −0.49755516345768189652383216562, 1.53806121446060628023533529611, 2.86744945854710397661093439013, 4.35946345999006345886377154069, 5.95282395093077537264666156068, 7.11591278863364332808680794027, 8.049811583207952341012810125258, 9.073693298670133913334588633181, 10.88370275859635664320674938527, 12.92702420718986965769525541507, 14.37887563081458558533149745904

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