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Properties

Label 2-3-3.2-c64-0-15
Degree $2$
Conductor $3$
Sign $-0.161 - 0.986i$
Analytic cond. $77.8210$
Root an. cond. $8.82162$
Motivic weight $64$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.55e9i·2-s + (−2.98e14 − 1.82e15i)3-s − 3.86e19·4-s + 3.78e22i·5-s + (−1.38e25 + 2.25e24i)6-s + 1.61e27·7-s + 1.53e29i·8-s + (−3.25e30 + 1.09e30i)9-s + 2.86e32·10-s − 1.51e33i·11-s + (1.15e34 + 7.07e34i)12-s − 1.60e35·13-s − 1.22e37i·14-s + (6.92e37 − 1.13e37i)15-s + 4.43e38·16-s − 1.42e39i·17-s + ⋯
L(s)  = 1  − 1.76i·2-s + (−0.161 − 0.986i)3-s − 2.09·4-s + 1.62i·5-s + (−1.73 + 0.283i)6-s + 1.46·7-s + 1.93i·8-s + (−0.948 + 0.318i)9-s + 2.86·10-s − 0.717i·11-s + (0.338 + 2.07i)12-s − 0.361·13-s − 2.57i·14-s + (1.60 − 0.262i)15-s + 1.30·16-s − 0.601i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(65-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+32) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.161 - 0.986i$
Analytic conductor: \(77.8210\)
Root analytic conductor: \(8.82162\)
Motivic weight: \(64\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :32),\ -0.161 - 0.986i)\)

Particular Values

\(L(\frac{65}{2})\) \(\approx\) \(1.008842508\)
\(L(\frac12)\) \(\approx\) \(1.008842508\)
\(L(33)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.98e14 + 1.82e15i)T \)
good2 \( 1 + 7.55e9iT - 1.84e19T^{2} \)
5 \( 1 - 3.78e22iT - 5.42e44T^{2} \)
7 \( 1 - 1.61e27T + 1.21e54T^{2} \)
11 \( 1 + 1.51e33iT - 4.45e66T^{2} \)
13 \( 1 + 1.60e35T + 1.96e71T^{2} \)
17 \( 1 + 1.42e39iT - 5.60e78T^{2} \)
19 \( 1 - 7.62e40T + 6.92e81T^{2} \)
23 \( 1 + 3.71e43iT - 1.41e87T^{2} \)
29 \( 1 - 6.21e46iT - 3.92e93T^{2} \)
31 \( 1 - 5.53e47T + 2.79e95T^{2} \)
37 \( 1 + 2.20e50T + 2.31e100T^{2} \)
41 \( 1 + 4.47e51iT - 1.65e103T^{2} \)
43 \( 1 + 1.29e52T + 3.48e104T^{2} \)
47 \( 1 + 1.54e53iT - 1.03e107T^{2} \)
53 \( 1 + 2.62e55iT - 2.25e110T^{2} \)
59 \( 1 + 6.43e55iT - 2.16e113T^{2} \)
61 \( 1 + 8.05e56T + 1.82e114T^{2} \)
67 \( 1 - 1.17e58T + 7.39e116T^{2} \)
71 \( 1 + 6.05e58iT - 3.02e118T^{2} \)
73 \( 1 + 5.83e59T + 1.78e119T^{2} \)
79 \( 1 + 3.50e60T + 2.80e121T^{2} \)
83 \( 1 - 5.79e60iT - 6.62e122T^{2} \)
89 \( 1 - 5.84e61iT - 5.76e124T^{2} \)
97 \( 1 + 1.93e63T + 1.42e127T^{2} \)
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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

−11.97837077330569461639465597421, −11.27473718205728632520782260280, −10.42649913746559595649670363372, −8.440223071822364112194357539246, −7.02188528579670795110302138352, −5.13563222598502353498884448106, −3.29300918774283795411199782111, −2.44011455851131658737004747224, −1.48375844900301844354014213843, −0.25778578096580290797123514566, 1.27675396795341913321111340515, 4.33696264944431243626860135635, 4.87241895815030976606016356482, 5.67410306278271851456188184289, 7.75535946664517760020764391940, 8.591119755575776760729428878329, 9.661796014113585072306136223271, 11.89754582421563739977126557492, 13.74152816907005081488467324309, 15.01512792797598777002687721272

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