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Properties

Label 2-51-3.2-c2-0-9
Degree $2$
Conductor $51$
Sign $-0.903 + 0.429i$
Analytic cond. $1.38964$
Root an. cond. $1.17883$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.98i·2-s + (−1.28 − 2.70i)3-s − 4.91·4-s + 2.35i·5-s + (−8.08 + 3.84i)6-s + 5.84·7-s + 2.73i·8-s + (−5.68 + 6.98i)9-s + 7.04·10-s − 13.8i·11-s + (6.33 + 13.3i)12-s + 6.24·13-s − 17.4i·14-s + (6.39 − 3.03i)15-s − 11.5·16-s + 4.12i·17-s + ⋯
L(s)  = 1  − 1.49i·2-s + (−0.429 − 0.903i)3-s − 1.22·4-s + 0.471i·5-s + (−1.34 + 0.641i)6-s + 0.835·7-s + 0.341i·8-s + (−0.631 + 0.775i)9-s + 0.704·10-s − 1.25i·11-s + (0.527 + 1.10i)12-s + 0.480·13-s − 1.24i·14-s + (0.426 − 0.202i)15-s − 0.719·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.903 + 0.429i$
Analytic conductor: \(1.38964\)
Root analytic conductor: \(1.17883\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1),\ -0.903 + 0.429i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.225640 - 0.999903i\)
\(L(\frac12)\) \(\approx\) \(0.225640 - 0.999903i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 + 2.70i)T \)
17 \( 1 - 4.12iT \)
good2 \( 1 + 2.98iT - 4T^{2} \)
5 \( 1 - 2.35iT - 25T^{2} \)
7 \( 1 - 5.84T + 49T^{2} \)
11 \( 1 + 13.8iT - 121T^{2} \)
13 \( 1 - 6.24T + 169T^{2} \)
19 \( 1 - 27.7T + 361T^{2} \)
23 \( 1 - 36.5iT - 529T^{2} \)
29 \( 1 - 6.36iT - 841T^{2} \)
31 \( 1 + 33.3T + 961T^{2} \)
37 \( 1 + 59.7T + 1.36e3T^{2} \)
41 \( 1 + 19.5iT - 1.68e3T^{2} \)
43 \( 1 - 65.4T + 1.84e3T^{2} \)
47 \( 1 + 9.62iT - 2.20e3T^{2} \)
53 \( 1 - 56.9iT - 2.80e3T^{2} \)
59 \( 1 - 45.4iT - 3.48e3T^{2} \)
61 \( 1 + 22.1T + 3.72e3T^{2} \)
67 \( 1 + 23.8T + 4.48e3T^{2} \)
71 \( 1 + 17.5iT - 5.04e3T^{2} \)
73 \( 1 - 18.8T + 5.32e3T^{2} \)
79 \( 1 + 51.4T + 6.24e3T^{2} \)
83 \( 1 + 102. iT - 6.88e3T^{2} \)
89 \( 1 + 59.0iT - 7.92e3T^{2} \)
97 \( 1 + 155.T + 9.40e3T^{2} \)
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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

−14.11395059220785950329646829667, −13.46290583207262307600198736423, −12.14859482240248275834913388806, −11.26937410775808686138494735886, −10.73280632907235504469448406853, −8.905870127625954072197633930529, −7.37781042987950898322933045528, −5.55183244224686435418225412983, −3.25340433496596173516261994536, −1.37049279089405681322086723441, 4.56383454603194044106044100370, 5.38972794906736752100620265429, 6.97582764643822361547886462207, 8.376017068008688964853845026077, 9.475794356396557180759080914151, 11.00747051435825031378005255994, 12.36627656517608557393411028755, 14.14815629359355160560243620634, 14.87180354917030045371471114033, 15.86975337513320207477038384966

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