Properties

Label 2-21-21.17-c35-0-66
Degree $2$
Conductor $21$
Sign $-0.878 + 0.477i$
Analytic cond. $162.949$
Root an. cond. $12.7651$
Motivic weight $35$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93e8 − 1.11e8i)3-s + (−1.71e10 + 2.97e10i)4-s + (−2.23e14 − 5.73e14i)7-s + (2.50e16 + 4.33e16i)9-s + (6.65e18 − 3.84e18i)12-s − 5.01e19i·13-s + (−5.90e20 − 1.02e21i)16-s + (−1.45e22 + 8.38e21i)19-s + (−2.08e22 + 1.36e23i)21-s + (1.45e24 − 2.52e24i)25-s + (−1.07e9 − 1.11e25i)27-s + (2.09e25 + 3.21e24i)28-s + (4.96e25 + 2.86e25i)31-s + (−1.71e27 + 1.37e11i)36-s + (7.17e26 + 1.24e27i)37-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (−0.362 − 0.931i)7-s + (0.500 + 0.866i)9-s + (0.866 − 0.499i)12-s − 1.60i·13-s + (−0.499 − 0.866i)16-s + (−0.607 + 0.350i)19-s + (−0.151 + 0.988i)21-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (0.988 + 0.151i)28-s + (0.395 + 0.228i)31-s − 36-s + (0.258 + 0.447i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.878 + 0.477i$
Analytic conductor: \(162.949\)
Root analytic conductor: \(12.7651\)
Motivic weight: \(35\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :35/2),\ -0.878 + 0.477i)\)

Particular Values

\(L(18)\) \(\approx\) \(0.7275962579\)
\(L(\frac12)\) \(\approx\) \(0.7275962579\)
\(L(\frac{37}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.93e8 + 1.11e8i)T \)
7 \( 1 + (2.23e14 + 5.73e14i)T \)
good2 \( 1 + (1.71e10 - 2.97e10i)T^{2} \)
5 \( 1 + (-1.45e24 + 2.52e24i)T^{2} \)
11 \( 1 + (1.40e36 + 2.43e36i)T^{2} \)
13 \( 1 + 5.01e19iT - 9.72e38T^{2} \)
17 \( 1 + (-5.81e42 - 1.00e43i)T^{2} \)
19 \( 1 + (1.45e22 - 8.38e21i)T + (2.85e44 - 4.94e44i)T^{2} \)
23 \( 1 + (2.28e47 - 3.96e47i)T^{2} \)
29 \( 1 - 1.52e51T^{2} \)
31 \( 1 + (-4.96e25 - 2.86e25i)T + (7.88e51 + 1.36e52i)T^{2} \)
37 \( 1 + (-7.17e26 - 1.24e27i)T + (-3.85e54 + 6.67e54i)T^{2} \)
41 \( 1 + 2.80e56T^{2} \)
43 \( 1 - 6.94e28T + 1.48e57T^{2} \)
47 \( 1 + (-1.66e58 + 2.89e58i)T^{2} \)
53 \( 1 + (1.11e60 + 1.93e60i)T^{2} \)
59 \( 1 + (-4.77e61 - 8.26e61i)T^{2} \)
61 \( 1 + (-1.11e31 + 6.42e30i)T + (1.53e62 - 2.65e62i)T^{2} \)
67 \( 1 + (-5.51e31 + 9.55e31i)T + (-4.08e63 - 7.08e63i)T^{2} \)
71 \( 1 - 6.22e64T^{2} \)
73 \( 1 + (-4.16e32 - 2.40e32i)T + (8.22e64 + 1.42e65i)T^{2} \)
79 \( 1 + (-6.79e32 - 1.17e33i)T + (-1.30e66 + 2.26e66i)T^{2} \)
83 \( 1 + 1.47e67T^{2} \)
89 \( 1 + (-8.46e67 + 1.46e68i)T^{2} \)
97 \( 1 + 9.65e34iT - 3.44e69T^{2} \)
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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

−10.95323708837163594021090103007, −10.01359475385188510441448479298, −8.260511658592898329462771415867, −7.45741552922853064823835045939, −6.30330192552762989897248401937, −4.96841081845382692804766035282, −3.89181966573401014847758825833, −2.65047310455463023216562536954, −0.899881442385769637931273516010, −0.24345685328135260614960632439, 0.942008023110867236384229169918, 2.18529138589837449068738895129, 3.97791156693372102060613351295, 4.92184650648417052024467218192, 5.91231097111558047840653327116, 6.74807217142832369620005303356, 8.970623294105340148626751275444, 9.497922675750068443263982667116, 10.77821656325845291539486385410, 11.73483920741757462634124708152

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