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Properties

Label 2-21-21.20-c21-0-35
Degree $2$
Conductor $21$
Sign $0.918 - 0.396i$
Analytic cond. $58.6902$
Root an. cond. $7.66095$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 367. i·2-s + (9.13e4 + 4.60e4i)3-s + 1.96e6·4-s + 3.23e7·5-s + (1.69e7 − 3.35e7i)6-s + (−4.79e8 + 5.73e8i)7-s − 1.49e9i·8-s + (6.22e9 + 8.40e9i)9-s − 1.18e10i·10-s − 1.31e11i·11-s + (1.79e11 + 9.03e10i)12-s + 6.69e11i·13-s + (2.10e11 + 1.75e11i)14-s + (2.95e12 + 1.48e12i)15-s + 3.56e12·16-s + 9.54e12·17-s + ⋯
L(s)  = 1  − 0.253i·2-s + (0.892 + 0.450i)3-s + 0.935·4-s + 1.48·5-s + (0.114 − 0.226i)6-s + (−0.641 + 0.767i)7-s − 0.490i·8-s + (0.594 + 0.803i)9-s − 0.375i·10-s − 1.52i·11-s + (0.835 + 0.421i)12-s + 1.34i·13-s + (0.194 + 0.162i)14-s + (1.32 + 0.666i)15-s + 0.811·16-s + 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.918 - 0.396i$
Analytic conductor: \(58.6902\)
Root analytic conductor: \(7.66095\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :21/2),\ 0.918 - 0.396i)\)

Particular Values

\(L(11)\) \(\approx\) \(5.029585890\)
\(L(\frac12)\) \(\approx\) \(5.029585890\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.13e4 - 4.60e4i)T \)
7 \( 1 + (4.79e8 - 5.73e8i)T \)
good2 \( 1 + 367. iT - 2.09e6T^{2} \)
5 \( 1 - 3.23e7T + 4.76e14T^{2} \)
11 \( 1 + 1.31e11iT - 7.40e21T^{2} \)
13 \( 1 - 6.69e11iT - 2.47e23T^{2} \)
17 \( 1 - 9.54e12T + 6.90e25T^{2} \)
19 \( 1 - 1.11e13iT - 7.14e26T^{2} \)
23 \( 1 + 2.12e14iT - 3.94e28T^{2} \)
29 \( 1 + 1.26e15iT - 5.13e30T^{2} \)
31 \( 1 - 5.66e15iT - 2.08e31T^{2} \)
37 \( 1 + 4.08e16T + 8.55e32T^{2} \)
41 \( 1 + 2.27e16T + 7.38e33T^{2} \)
43 \( 1 - 1.15e17T + 2.00e34T^{2} \)
47 \( 1 + 1.81e17T + 1.30e35T^{2} \)
53 \( 1 - 7.35e17iT - 1.62e36T^{2} \)
59 \( 1 - 2.94e18T + 1.54e37T^{2} \)
61 \( 1 + 1.08e18iT - 3.10e37T^{2} \)
67 \( 1 + 6.78e18T + 2.22e38T^{2} \)
71 \( 1 - 2.06e19iT - 7.52e38T^{2} \)
73 \( 1 + 4.03e19iT - 1.34e39T^{2} \)
79 \( 1 - 9.41e19T + 7.08e39T^{2} \)
83 \( 1 - 2.74e19T + 1.99e40T^{2} \)
89 \( 1 + 4.43e20T + 8.65e40T^{2} \)
97 \( 1 - 8.62e20iT - 5.27e41T^{2} \)
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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

−13.68995550272825648584775282421, −12.25470259830762909140619792682, −10.61731291642779476954639374311, −9.634600770653356348789824479419, −8.585303693737112865794312444438, −6.61407977986878479362446660670, −5.60690987407345271760885342308, −3.38023206354061251101800851253, −2.46604871909020172558541621733, −1.48283537260570273140306996586, 1.20138009126360946893523276377, 2.17622385748803838257409677250, 3.28118011315050020021220367628, 5.56700955649146451985843817170, 6.85287826311048579595815482836, 7.68669538982983945278571317657, 9.645170808350891579239114986663, 10.26660486600820083715665778232, 12.46814087965536243994943161049, 13.34522537449287031186697939828

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