Properties

Label 2-21-21.17-c37-0-53
Degree $2$
Conductor $21$
Sign $0.811 + 0.584i$
Analytic cond. $182.099$
Root an. cond. $13.4944$
Motivic weight $37$
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 − 3.35e8i)3-s + (−6.87e10 + 1.19e11i)4-s + (2.94e15 − 3.14e15i)7-s + (2.25e17 + 3.89e17i)9-s + (7.98e19 − 4.61e19i)12-s + 3.52e20i·13-s + (−9.44e21 − 1.63e22i)16-s + (7.84e23 − 4.52e23i)19-s + (−2.76e24 + 8.42e23i)21-s + (3.63e25 − 6.30e25i)25-s + (1.71e10 − 3.02e26i)27-s + (1.72e26 + 5.66e26i)28-s + (6.74e27 + 3.89e27i)31-s + (−6.18e28 − 4.39e12i)36-s + (−9.92e27 − 1.71e28i)37-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (0.682 − 0.730i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + 0.869i·13-s + (−0.499 − 0.866i)16-s + (1.72 − 0.997i)19-s + (−0.956 + 0.291i)21-s + (0.5 − 0.866i)25-s − 1.00i·27-s + (0.291 + 0.956i)28-s + (1.73 + 0.999i)31-s − 0.999·36-s + (−0.0966 − 0.167i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(182.099\)
Root analytic conductor: \(13.4944\)
Motivic weight: \(37\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :37/2),\ 0.811 + 0.584i)\)

Particular Values

\(L(19)\) \(\approx\) \(1.512631105\)
\(L(\frac12)\) \(\approx\) \(1.512631105\)
\(L(\frac{39}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.81e8 + 3.35e8i)T \)
7 \( 1 + (-2.94e15 + 3.14e15i)T \)
good2 \( 1 + (6.87e10 - 1.19e11i)T^{2} \)
5 \( 1 + (-3.63e25 + 6.30e25i)T^{2} \)
11 \( 1 + (1.70e38 + 2.94e38i)T^{2} \)
13 \( 1 - 3.52e20iT - 1.64e41T^{2} \)
17 \( 1 + (-1.68e45 - 2.91e45i)T^{2} \)
19 \( 1 + (-7.84e23 + 4.52e23i)T + (1.03e47 - 1.78e47i)T^{2} \)
23 \( 1 + (1.21e50 - 2.09e50i)T^{2} \)
29 \( 1 - 1.28e54T^{2} \)
31 \( 1 + (-6.74e27 - 3.89e27i)T + (7.57e54 + 1.31e55i)T^{2} \)
37 \( 1 + (9.92e27 + 1.71e28i)T + (-5.27e57 + 9.14e57i)T^{2} \)
41 \( 1 + 4.70e59T^{2} \)
43 \( 1 + 3.12e30T + 2.74e60T^{2} \)
47 \( 1 + (-3.68e61 + 6.38e61i)T^{2} \)
53 \( 1 + (3.14e63 + 5.44e63i)T^{2} \)
59 \( 1 + (-1.66e65 - 2.87e65i)T^{2} \)
61 \( 1 + (1.77e33 - 1.02e33i)T + (5.70e65 - 9.87e65i)T^{2} \)
67 \( 1 + (5.13e33 - 8.89e33i)T + (-1.83e67 - 3.17e67i)T^{2} \)
71 \( 1 - 3.13e68T^{2} \)
73 \( 1 + (-5.10e34 - 2.94e34i)T + (4.38e68 + 7.59e68i)T^{2} \)
79 \( 1 + (1.11e35 + 1.93e35i)T + (-8.15e69 + 1.41e70i)T^{2} \)
83 \( 1 + 1.01e71T^{2} \)
89 \( 1 + (-6.70e71 + 1.16e72i)T^{2} \)
97 \( 1 + 6.37e36iT - 3.24e73T^{2} \)
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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

−11.46451520800762463843753189871, −10.09172469543090865734279668061, −8.596372921485277732339030703105, −7.48611697848609999479237306604, −6.72631571756274615824458733103, −5.01918170998941696182132812785, −4.40294455097480153695639171462, −2.92213030573735521262456048016, −1.41239151959645660431877604177, −0.50151550829269452294415441616, 0.72493513179751077437943767995, 1.55261749067565702897214463170, 3.30707521816141849256237759062, 4.78742299386941547732331703292, 5.36907821426381340439569147469, 6.24528152846207939646504847702, 7.971459471117542931949314887431, 9.362158840709763974524425570880, 10.17104326271434930368895140009, 11.26956508758484783160755901112

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