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Properties

Label 2-3-3.2-c70-0-16
Degree $2$
Conductor $3$
Sign $0.151 - 0.988i$
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  + 3.74e10i·2-s + (−7.56e15 + 4.94e16i)3-s − 2.23e20·4-s − 1.07e24i·5-s + (−1.85e27 − 2.83e26i)6-s + 6.73e29·7-s + 3.58e31i·8-s + (−2.38e33 − 7.47e32i)9-s + 4.02e34·10-s − 9.56e35i·11-s + (1.68e36 − 1.10e37i)12-s + 9.40e38·13-s + 2.52e40i·14-s + (5.31e40 + 8.12e39i)15-s − 1.60e42·16-s − 1.38e43i·17-s + ⋯
L(s)  = 1  + 1.09i·2-s + (−0.151 + 0.988i)3-s − 0.189·4-s − 0.369i·5-s + (−1.07 − 0.164i)6-s + 1.77·7-s + 0.884i·8-s + (−0.954 − 0.298i)9-s + 0.402·10-s − 0.340i·11-s + (0.0285 − 0.186i)12-s + 0.966·13-s + 1.93i·14-s + (0.364 + 0.0557i)15-s − 1.15·16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.151 - 0.988i$
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: \((2,\ 3,\ (\ :35),\ 0.151 - 0.988i)\)

Particular Values

\(L(\frac{71}{2})\) \(\approx\) \(3.019741923\)
\(L(\frac12)\) \(\approx\) \(3.019741923\)
\(L(36)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.56e15 - 4.94e16i)T \)
good2 \( 1 - 3.74e10iT - 1.18e21T^{2} \)
5 \( 1 + 1.07e24iT - 8.47e48T^{2} \)
7 \( 1 - 6.73e29T + 1.43e59T^{2} \)
11 \( 1 + 9.56e35iT - 7.89e72T^{2} \)
13 \( 1 - 9.40e38T + 9.46e77T^{2} \)
17 \( 1 + 1.38e43iT - 1.35e86T^{2} \)
19 \( 1 + 4.21e44T + 3.25e89T^{2} \)
23 \( 1 + 7.41e47iT - 2.09e95T^{2} \)
29 \( 1 + 6.56e50iT - 2.33e102T^{2} \)
31 \( 1 - 2.15e52T + 2.48e104T^{2} \)
37 \( 1 - 4.93e54T + 5.94e109T^{2} \)
41 \( 1 - 5.23e55iT - 7.85e112T^{2} \)
43 \( 1 - 1.48e57T + 2.20e114T^{2} \)
47 \( 1 + 1.27e58iT - 1.11e117T^{2} \)
53 \( 1 + 3.94e60iT - 5.00e120T^{2} \)
59 \( 1 + 9.57e61iT - 9.11e123T^{2} \)
61 \( 1 + 2.37e61T + 9.39e124T^{2} \)
67 \( 1 + 8.71e63T + 6.68e127T^{2} \)
71 \( 1 + 4.27e64iT - 3.87e129T^{2} \)
73 \( 1 - 1.50e65T + 2.70e130T^{2} \)
79 \( 1 + 2.81e66T + 6.82e132T^{2} \)
83 \( 1 + 1.40e66iT - 2.16e134T^{2} \)
89 \( 1 - 8.71e67iT - 2.86e136T^{2} \)
97 \( 1 + 5.94e69T + 1.18e139T^{2} \)
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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

−14.05344086411252730807444091693, −11.60206000272563633955122060111, −10.78150939568833743957437502322, −8.735776127047022254453309874162, −8.123883035045591501259519971097, −6.35805213740660099744861622096, −5.09788506412877941875475666140, −4.45059907015026820009746604349, −2.45561123325711016367799402976, −0.74980424145027776460087582778, 1.19325894102919511950457936461, 1.56145889530364022457106557457, 2.71018966995499253192309596806, 4.25273968405749688605031349324, 5.97142487375902706706662540662, 7.38814172544820081603848536385, 8.562080491860048862983991298328, 10.71277841713680011779913197582, 11.31282757176774526252002124447, 12.42745087170852870526321243169

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