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Properties

Label 2-3-3.2-c70-0-13
Degree $2$
Conductor $3$
Sign $-0.694 - 0.719i$
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  − 4.22e9i·2-s + (3.47e16 + 3.60e16i)3-s + 1.16e21·4-s + 5.48e24i·5-s + (1.52e26 − 1.46e26i)6-s + 4.84e29·7-s − 9.89e30i·8-s + (−9.14e31 + 2.50e33i)9-s + 2.31e34·10-s + 3.11e35i·11-s + (4.03e37 + 4.18e37i)12-s − 1.13e39·13-s − 2.04e39i·14-s + (−1.97e41 + 1.90e41i)15-s + 1.33e42·16-s + 1.49e42i·17-s + ⋯
L(s)  = 1  − 0.122i·2-s + (0.694 + 0.719i)3-s + 0.984·4-s + 1.88i·5-s + (0.0884 − 0.0852i)6-s + 1.27·7-s − 0.243i·8-s + (−0.0365 + 0.999i)9-s + 0.231·10-s + 0.110i·11-s + (0.683 + 0.709i)12-s − 1.17·13-s − 0.157i·14-s + (−1.35 + 1.30i)15-s + 0.954·16-s + 0.128i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.694 - 0.719i$
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: \((2,\ 3,\ (\ :35),\ -0.694 - 0.719i)\)

Particular Values

\(L(\frac{71}{2})\) \(\approx\) \(4.313593829\)
\(L(\frac12)\) \(\approx\) \(4.313593829\)
\(L(36)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.47e16 - 3.60e16i)T \)
good2 \( 1 + 4.22e9iT - 1.18e21T^{2} \)
5 \( 1 - 5.48e24iT - 8.47e48T^{2} \)
7 \( 1 - 4.84e29T + 1.43e59T^{2} \)
11 \( 1 - 3.11e35iT - 7.89e72T^{2} \)
13 \( 1 + 1.13e39T + 9.46e77T^{2} \)
17 \( 1 - 1.49e42iT - 1.35e86T^{2} \)
19 \( 1 - 8.25e44T + 3.25e89T^{2} \)
23 \( 1 - 2.23e47iT - 2.09e95T^{2} \)
29 \( 1 - 1.79e51iT - 2.33e102T^{2} \)
31 \( 1 - 1.76e51T + 2.48e104T^{2} \)
37 \( 1 - 1.37e54T + 5.94e109T^{2} \)
41 \( 1 + 2.77e56iT - 7.85e112T^{2} \)
43 \( 1 - 1.44e57T + 2.20e114T^{2} \)
47 \( 1 + 3.31e57iT - 1.11e117T^{2} \)
53 \( 1 + 3.39e60iT - 5.00e120T^{2} \)
59 \( 1 + 1.24e62iT - 9.11e123T^{2} \)
61 \( 1 - 1.10e62T + 9.39e124T^{2} \)
67 \( 1 + 4.15e63T + 6.68e127T^{2} \)
71 \( 1 - 8.25e64iT - 3.87e129T^{2} \)
73 \( 1 - 2.28e64T + 2.70e130T^{2} \)
79 \( 1 - 9.17e65T + 6.82e132T^{2} \)
83 \( 1 + 1.81e67iT - 2.16e134T^{2} \)
89 \( 1 + 9.16e67iT - 2.86e136T^{2} \)
97 \( 1 + 4.22e69T + 1.18e139T^{2} \)
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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

−14.27018727738284818486807971982, −11.63514370537754173864536906564, −10.81520297687826067781123319316, −9.864837944407194325123070235089, −7.75047727438122714975005385301, −7.10733513908783539025142521464, −5.29938025831198582578562835448, −3.55350845213985768555030935929, −2.64765926572796620055138893362, −1.82272204536728996787587218796, 0.804227337641077085968486052963, 1.50595924071896516103972927258, 2.55171443544354125668043720312, 4.49763333840544218683598382346, 5.65937124237970461215364459770, 7.52750094608571823202291636914, 8.139263315368752147302392633340, 9.454017545357003300239919679467, 11.70379304241313321775117031946, 12.36945602453856589708270009373

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