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Properties

Label 2-3-3.2-c70-0-3
Degree $2$
Conductor $3$
Sign $0.985 + 0.169i$
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  − 4.67e10i·2-s + (−4.93e16 − 8.50e15i)3-s − 1.00e21·4-s + 4.28e24i·5-s + (−3.97e26 + 2.30e27i)6-s − 1.75e29·7-s − 8.16e30i·8-s + (2.35e33 + 8.38e32i)9-s + 2.00e35·10-s − 2.13e36i·11-s + (4.96e37 + 8.55e36i)12-s − 1.41e39·13-s + 8.21e39i·14-s + (3.64e40 − 2.11e41i)15-s − 1.56e42·16-s − 1.47e43i·17-s + ⋯
L(s)  = 1  − 1.36i·2-s + (−0.985 − 0.169i)3-s − 0.852·4-s + 1.47i·5-s + (−0.231 + 1.34i)6-s − 0.463·7-s − 0.201i·8-s + (0.942 + 0.334i)9-s + 2.00·10-s − 0.758i·11-s + (0.839 + 0.144i)12-s − 1.45·13-s + 0.630i·14-s + (0.250 − 1.45i)15-s − 1.12·16-s − 1.27i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.985 + 0.169i$
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: \((2,\ 3,\ (\ :35),\ 0.985 + 0.169i)\)

Particular Values

\(L(\frac{71}{2})\) \(\approx\) \(0.4418918309\)
\(L(\frac12)\) \(\approx\) \(0.4418918309\)
\(L(36)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.93e16 + 8.50e15i)T \)
good2 \( 1 + 4.67e10iT - 1.18e21T^{2} \)
5 \( 1 - 4.28e24iT - 8.47e48T^{2} \)
7 \( 1 + 1.75e29T + 1.43e59T^{2} \)
11 \( 1 + 2.13e36iT - 7.89e72T^{2} \)
13 \( 1 + 1.41e39T + 9.46e77T^{2} \)
17 \( 1 + 1.47e43iT - 1.35e86T^{2} \)
19 \( 1 + 5.25e44T + 3.25e89T^{2} \)
23 \( 1 + 2.67e47iT - 2.09e95T^{2} \)
29 \( 1 + 1.32e51iT - 2.33e102T^{2} \)
31 \( 1 - 2.33e52T + 2.48e104T^{2} \)
37 \( 1 + 9.09e54T + 5.94e109T^{2} \)
41 \( 1 + 1.39e56iT - 7.85e112T^{2} \)
43 \( 1 - 1.81e57T + 2.20e114T^{2} \)
47 \( 1 - 3.74e58iT - 1.11e117T^{2} \)
53 \( 1 - 1.66e60iT - 5.00e120T^{2} \)
59 \( 1 + 1.05e62iT - 9.11e123T^{2} \)
61 \( 1 + 2.28e62T + 9.39e124T^{2} \)
67 \( 1 + 1.70e63T + 6.68e127T^{2} \)
71 \( 1 - 3.89e64iT - 3.87e129T^{2} \)
73 \( 1 + 3.18e64T + 2.70e130T^{2} \)
79 \( 1 + 4.58e66T + 6.82e132T^{2} \)
83 \( 1 - 7.08e66iT - 2.16e134T^{2} \)
89 \( 1 - 2.22e68iT - 2.86e136T^{2} \)
97 \( 1 - 4.55e69T + 1.18e139T^{2} \)
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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

−12.46869296252113724681115696842, −11.48696604096073735337048951453, −10.58444916098383907330574038345, −9.764362191632119401846225805385, −7.20677123593137991116810064598, −6.28501751709841966609725667624, −4.49874704709884210071752186863, −2.99929199588660383701646980630, −2.29702873400894913898060429606, −0.60588117942516907603900564035, 0.18437698104493462034866024652, 1.72220038318413873492385333913, 4.41847391659037067460456518388, 5.08357078084084578716055892105, 6.18899672537401654912630594023, 7.35777958933324116731515353044, 8.749004666437603396235830204769, 10.07674670981020257336038790121, 12.09758047202798109724372364258, 12.94561749365138742951810318228

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