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Properties

Label 2-3-3.2-c70-0-9
Degree $2$
Conductor $3$
Sign $0.844 - 0.536i$
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  − 1.12e10i·2-s + (−4.22e16 + 2.68e16i)3-s + 1.05e21·4-s + 2.30e23i·5-s + (3.01e26 + 4.75e26i)6-s + 8.16e28·7-s − 2.51e31i·8-s + (1.06e33 − 2.26e33i)9-s + 2.58e33·10-s + 4.04e36i·11-s + (−4.45e37 + 2.82e37i)12-s + 4.37e38·13-s − 9.17e38i·14-s + (−6.17e39 − 9.72e39i)15-s + 9.61e41·16-s − 1.30e42i·17-s + ⋯
L(s)  = 1  − 0.327i·2-s + (−0.844 + 0.536i)3-s + 0.892·4-s + 0.0790i·5-s + (0.175 + 0.276i)6-s + 0.215·7-s − 0.619i·8-s + (0.425 − 0.905i)9-s + 0.0258·10-s + 1.44i·11-s + (−0.753 + 0.478i)12-s + 0.449·13-s − 0.0705i·14-s + (−0.0424 − 0.0667i)15-s + 0.689·16-s − 0.112i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.536i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (0.844 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.844 - 0.536i$
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: \((2,\ 3,\ (\ :35),\ 0.844 - 0.536i)\)

Particular Values

\(L(\frac{71}{2})\) \(\approx\) \(2.255876330\)
\(L(\frac12)\) \(\approx\) \(2.255876330\)
\(L(36)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.22e16 - 2.68e16i)T \)
good2 \( 1 + 1.12e10iT - 1.18e21T^{2} \)
5 \( 1 - 2.30e23iT - 8.47e48T^{2} \)
7 \( 1 - 8.16e28T + 1.43e59T^{2} \)
11 \( 1 - 4.04e36iT - 7.89e72T^{2} \)
13 \( 1 - 4.37e38T + 9.46e77T^{2} \)
17 \( 1 + 1.30e42iT - 1.35e86T^{2} \)
19 \( 1 - 2.51e44T + 3.25e89T^{2} \)
23 \( 1 - 4.75e46iT - 2.09e95T^{2} \)
29 \( 1 + 1.88e51iT - 2.33e102T^{2} \)
31 \( 1 + 1.77e52T + 2.48e104T^{2} \)
37 \( 1 + 1.27e54T + 5.94e109T^{2} \)
41 \( 1 - 1.52e56iT - 7.85e112T^{2} \)
43 \( 1 - 6.99e56T + 2.20e114T^{2} \)
47 \( 1 + 4.05e58iT - 1.11e117T^{2} \)
53 \( 1 - 4.34e60iT - 5.00e120T^{2} \)
59 \( 1 + 1.21e62iT - 9.11e123T^{2} \)
61 \( 1 - 1.58e62T + 9.39e124T^{2} \)
67 \( 1 - 1.73e63T + 6.68e127T^{2} \)
71 \( 1 - 5.86e63iT - 3.87e129T^{2} \)
73 \( 1 - 9.24e64T + 2.70e130T^{2} \)
79 \( 1 - 3.24e66T + 6.82e132T^{2} \)
83 \( 1 + 5.01e66iT - 2.16e134T^{2} \)
89 \( 1 - 2.40e68iT - 2.86e136T^{2} \)
97 \( 1 - 4.56e69T + 1.18e139T^{2} \)
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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

−12.66531956567218678085418415990, −11.64864620081648224573015900460, −10.62567050250541375287880089947, −9.557628246781098224539690485476, −7.40775018531613803119385247085, −6.32833818987077315194578271302, −4.94960086074795507080164416441, −3.63844085592955582850239471189, −2.10736180867332434707019310822, −0.915370530590945245954824148641, 0.68919501733080980184552427006, 1.71045669651024568318319333116, 3.21642407729274514815098087470, 5.23484239273363660930194068844, 6.14363692156478827183532482192, 7.21671018001611869268670635166, 8.479714155942264223793599003901, 10.74473734407713000839306357083, 11.39469877741581699487650402092, 12.75627226550248746790959205026

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