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Properties

Label 2-3-3.2-c70-0-11
Degree $2$
Conductor $3$
Sign $0.540 - 0.841i$
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  + 3.59e10i·2-s + (−2.70e16 + 4.20e16i)3-s − 1.11e20·4-s + 4.52e24i·5-s + (−1.51e27 − 9.72e26i)6-s − 7.40e29·7-s + 3.84e31i·8-s + (−1.03e33 − 2.27e33i)9-s − 1.62e35·10-s − 3.10e36i·11-s + (3.00e36 − 4.67e36i)12-s + 2.12e37·13-s − 2.66e40i·14-s + (−1.90e41 − 1.22e41i)15-s − 1.51e42·16-s − 6.65e41i·17-s + ⋯
L(s)  = 1  + 1.04i·2-s + (−0.540 + 0.841i)3-s − 0.0941·4-s + 1.55i·5-s + (−0.879 − 0.565i)6-s − 1.95·7-s + 0.947i·8-s + (−0.415 − 0.909i)9-s − 1.62·10-s − 1.10i·11-s + (0.0509 − 0.0792i)12-s + 0.0218·13-s − 2.04i·14-s + (−1.30 − 0.839i)15-s − 1.08·16-s − 0.0572i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.540 - 0.841i$
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: \((2,\ 3,\ (\ :35),\ 0.540 - 0.841i)\)

Particular Values

\(L(\frac{71}{2})\) \(\approx\) \(0.3215189789\)
\(L(\frac12)\) \(\approx\) \(0.3215189789\)
\(L(36)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.70e16 - 4.20e16i)T \)
good2 \( 1 - 3.59e10iT - 1.18e21T^{2} \)
5 \( 1 - 4.52e24iT - 8.47e48T^{2} \)
7 \( 1 + 7.40e29T + 1.43e59T^{2} \)
11 \( 1 + 3.10e36iT - 7.89e72T^{2} \)
13 \( 1 - 2.12e37T + 9.46e77T^{2} \)
17 \( 1 + 6.65e41iT - 1.35e86T^{2} \)
19 \( 1 - 3.67e44T + 3.25e89T^{2} \)
23 \( 1 - 5.00e47iT - 2.09e95T^{2} \)
29 \( 1 + 4.46e50iT - 2.33e102T^{2} \)
31 \( 1 + 1.45e52T + 2.48e104T^{2} \)
37 \( 1 - 7.83e54T + 5.94e109T^{2} \)
41 \( 1 + 1.04e56iT - 7.85e112T^{2} \)
43 \( 1 + 1.18e57T + 2.20e114T^{2} \)
47 \( 1 + 1.39e58iT - 1.11e117T^{2} \)
53 \( 1 + 1.97e59iT - 5.00e120T^{2} \)
59 \( 1 - 1.24e61iT - 9.11e123T^{2} \)
61 \( 1 + 5.47e62T + 9.39e124T^{2} \)
67 \( 1 + 4.78e63T + 6.68e127T^{2} \)
71 \( 1 + 8.14e64iT - 3.87e129T^{2} \)
73 \( 1 + 2.23e65T + 2.70e130T^{2} \)
79 \( 1 + 1.91e66T + 6.82e132T^{2} \)
83 \( 1 - 1.95e67iT - 2.16e134T^{2} \)
89 \( 1 - 5.75e67iT - 2.86e136T^{2} \)
97 \( 1 - 1.88e69T + 1.18e139T^{2} \)
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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

−13.72104766617913496391323393593, −11.55379339576316273945451214453, −10.48150502289711345638060040337, −9.299538125757631312448095575525, −7.26980481549418312207627543972, −6.29416611164668009609402030309, −5.75230888234308975980334323900, −3.51084253863534241726165020085, −2.91201468699434050460755070047, −0.10976250185132912382823977624, 0.70286754256336803689480548483, 1.68851787010348647897882516376, 2.93144819562480023411459476581, 4.47456814430832548029022866358, 6.04202679692094522419015591285, 7.20945063828128805998967481435, 9.140189763991270366443246994415, 10.16304774895500783506128580460, 11.93389795129739212796729439517, 12.75198411367753218872950264292

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