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Properties

Label 2-21-21.5-c23-0-35
Degree $2$
Conductor $21$
Sign $0.211 + 0.977i$
Analytic cond. $70.3928$
Root an. cond. $8.39004$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.65e5 + 1.53e5i)3-s + (−4.19e6 − 7.26e6i)4-s + (4.93e9 − 1.74e9i)7-s + (4.70e10 − 8.15e10i)9-s + (2.22e12 + 1.28e12i)12-s − 8.36e12i·13-s + (−3.51e13 + 6.09e13i)16-s + (7.49e14 + 4.32e14i)19-s + (−1.04e15 + 1.21e15i)21-s + (5.96e15 + 1.03e16i)25-s + (2 + 2.88e16i)27-s + (−3.33e16 − 2.85e16i)28-s + (−2.44e17 + 1.41e17i)31-s − 7.89e17·36-s + (1.02e18 − 1.77e18i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.500i)3-s + (−0.5 − 0.866i)4-s + (0.942 − 0.333i)7-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)12-s − 1.29i·13-s + (−0.499 + 0.866i)16-s + (1.47 + 0.852i)19-s + (−0.650 + 0.759i)21-s + (0.500 + 0.866i)25-s + 0.999i·27-s + (−0.759 − 0.650i)28-s + (−1.72 + 0.997i)31-s − 0.999·36-s + (0.948 − 1.64i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(70.3928\)
Root analytic conductor: \(8.39004\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :23/2),\ 0.211 + 0.977i)\)

Particular Values

\(L(12)\) \(\approx\) \(1.407141595\)
\(L(\frac12)\) \(\approx\) \(1.407141595\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.65e5 - 1.53e5i)T \)
7 \( 1 + (-4.93e9 + 1.74e9i)T \)
good2 \( 1 + (4.19e6 + 7.26e6i)T^{2} \)
5 \( 1 + (-5.96e15 - 1.03e16i)T^{2} \)
11 \( 1 + (4.47e23 - 7.75e23i)T^{2} \)
13 \( 1 + 8.36e12iT - 4.17e25T^{2} \)
17 \( 1 + (-9.98e27 + 1.72e28i)T^{2} \)
19 \( 1 + (-7.49e14 - 4.32e14i)T + (1.28e29 + 2.23e29i)T^{2} \)
23 \( 1 + (1.04e31 + 1.80e31i)T^{2} \)
29 \( 1 - 4.31e33T^{2} \)
31 \( 1 + (2.44e17 - 1.41e17i)T + (1.00e34 - 1.73e34i)T^{2} \)
37 \( 1 + (-1.02e18 + 1.77e18i)T + (-5.85e35 - 1.01e36i)T^{2} \)
41 \( 1 + 1.24e37T^{2} \)
43 \( 1 - 5.06e18T + 3.71e37T^{2} \)
47 \( 1 + (-1.43e38 - 2.48e38i)T^{2} \)
53 \( 1 + (2.27e39 - 3.94e39i)T^{2} \)
59 \( 1 + (-2.68e40 + 4.64e40i)T^{2} \)
61 \( 1 + (-5.35e20 - 3.09e20i)T + (5.77e40 + 1.00e41i)T^{2} \)
67 \( 1 + (9.45e20 + 1.63e21i)T + (-4.99e41 + 8.65e41i)T^{2} \)
71 \( 1 - 3.79e42T^{2} \)
73 \( 1 + (-4.45e21 + 2.57e21i)T + (3.59e42 - 6.22e42i)T^{2} \)
79 \( 1 + (1.05e20 - 1.82e20i)T + (-2.21e43 - 3.82e43i)T^{2} \)
83 \( 1 + 1.37e44T^{2} \)
89 \( 1 + (-3.42e44 - 5.93e44i)T^{2} \)
97 \( 1 + 1.10e23iT - 4.96e45T^{2} \)
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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

−12.69384567445292265171837487965, −11.16148758290273052284797810705, −10.41672861414817540722457325835, −9.222832300599903201784964751967, −7.49224354736094772146058847012, −5.66014813535109076661070777147, −5.12364536369743120763734176788, −3.71990128892207607612322580791, −1.40314194551751284386810968994, −0.52437859922303641671903908166, 0.952983901044094813043617832249, 2.37168952741411211504883962441, 4.26561649546872621750489968457, 5.25960390992068595331298672927, 6.92519154502220729227694761671, 7.987984353548151344800905300434, 9.340750209581474426794812390089, 11.30177275228259221800933341925, 11.89691570558643866897033074777, 13.16694564848581433659405089647

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