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Properties

Label 2-21-21.11-c38-0-27
Degree $2$
Conductor $21$
Sign $0.937 - 0.347i$
Analytic cond. $192.073$
Root an. cond. $13.8590$
Motivic weight $38$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.81e8 − 1.00e9i)3-s + (−1.37e11 + 2.38e11i)4-s + (−6.70e15 + 9.21e15i)7-s + (−6.75e17 − 1.16e18i)9-s + (1.59e20 + 2.76e20i)12-s − 2.86e21·13-s + (−3.77e22 − 6.54e22i)16-s + (−1.27e24 − 2.21e24i)19-s + (5.37e24 + 1.21e25i)21-s + (−1.81e26 + 3.15e26i)25-s + (−1.57e27 + 1.37e11i)27-s + (−1.27e27 − 2.86e27i)28-s + (1.65e28 − 2.87e28i)31-s + 3.71e29·36-s + (−6.24e29 − 1.08e30i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.588 + 0.808i)7-s + (−0.500 − 0.866i)9-s + (0.5 + 0.866i)12-s − 1.95·13-s + (−0.499 − 0.866i)16-s + (−0.646 − 1.12i)19-s + (0.405 + 0.913i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (−0.405 − 0.913i)28-s + (0.765 − 1.32i)31-s + 1.00·36-s + (−0.999 − 1.73i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.937 - 0.347i$
Analytic conductor: \(192.073\)
Root analytic conductor: \(13.8590\)
Motivic weight: \(38\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :19),\ 0.937 - 0.347i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(0.7764597418\)
\(L(\frac12)\) \(\approx\) \(0.7764597418\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.81e8 + 1.00e9i)T \)
7 \( 1 + (6.70e15 - 9.21e15i)T \)
good2 \( 1 + (1.37e11 - 2.38e11i)T^{2} \)
5 \( 1 + (1.81e26 - 3.15e26i)T^{2} \)
11 \( 1 + (1.87e39 + 3.23e39i)T^{2} \)
13 \( 1 + 2.86e21T + 2.13e42T^{2} \)
17 \( 1 + (2.85e46 + 4.94e46i)T^{2} \)
19 \( 1 + (1.27e24 + 2.21e24i)T + (-1.95e48 + 3.38e48i)T^{2} \)
23 \( 1 + (2.78e51 - 4.82e51i)T^{2} \)
29 \( 1 - 3.72e55T^{2} \)
31 \( 1 + (-1.65e28 + 2.87e28i)T + (-2.34e56 - 4.06e56i)T^{2} \)
37 \( 1 + (6.24e29 + 1.08e30i)T + (-1.95e59 + 3.38e59i)T^{2} \)
41 \( 1 - 1.93e61T^{2} \)
43 \( 1 - 1.44e31T + 1.17e62T^{2} \)
47 \( 1 + (1.73e63 - 3.00e63i)T^{2} \)
53 \( 1 + (1.66e65 + 2.88e65i)T^{2} \)
59 \( 1 + (9.80e66 + 1.69e67i)T^{2} \)
61 \( 1 + (1.52e33 + 2.63e33i)T + (-3.47e67 + 6.02e67i)T^{2} \)
67 \( 1 + (4.77e34 - 8.26e34i)T + (-1.22e69 - 2.12e69i)T^{2} \)
71 \( 1 - 2.22e70T^{2} \)
73 \( 1 + (2.39e35 - 4.14e35i)T + (-3.20e70 - 5.54e70i)T^{2} \)
79 \( 1 + (-1.10e36 - 1.90e36i)T + (-6.43e71 + 1.11e72i)T^{2} \)
83 \( 1 - 8.41e72T^{2} \)
89 \( 1 + (5.96e73 - 1.03e74i)T^{2} \)
97 \( 1 + 2.18e37T + 3.14e75T^{2} \)
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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

−11.68656282945293843985838435032, −9.566409412547901375197320531422, −8.902282816556849691057469329704, −7.68614648887974239801070963346, −6.92778550447498053515056039871, −5.45857379868393793340062670222, −4.06043809879431966935653582558, −2.70588193408403579238082677011, −2.32117569178537157199230454261, −0.42937640956005989039006397452, 0.27660912583676661030245985425, 1.78361661522094495934479205683, 3.04712713904990880701969431492, 4.29852423094626223151791879661, 4.95636897620041493404662178539, 6.32834935927607390839177547962, 7.76346563571109002474958023071, 9.077251601684503824342081143864, 10.21163016893098419215768339786, 10.29957295485537133119845704223

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