Properties

Label 2-21-7.2-c29-0-30
Degree $2$
Conductor $21$
Sign $0.999 - 0.0256i$
Analytic cond. $111.883$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.50e3 + 1.64e4i)2-s + (2.39e6 + 4.14e6i)3-s + (8.76e7 + 1.51e8i)4-s + (4.01e9 − 6.95e9i)5-s − 9.09e10·6-s + (1.51e12 + 9.55e11i)7-s + (−1.35e13 + 0.000488i)8-s + (−1.14e13 + 1.98e13i)9-s + (7.62e13 + 1.32e14i)10-s + (−6.85e14 − 1.18e15i)11-s + (−4.19e14 + 7.26e14i)12-s − 8.39e14·13-s + (−3.01e16 + 1.59e16i)14-s + (3.83e16 − 2i)15-s + (8.16e16 − 1.41e17i)16-s + (−3.62e17 − 6.27e17i)17-s + ⋯
L(s)  = 1  + (−0.410 + 0.710i)2-s + (0.288 + 0.499i)3-s + (0.163 + 0.282i)4-s + (0.294 − 0.509i)5-s − 0.473·6-s + (0.846 + 0.532i)7-s − 1.08·8-s + (−0.166 + 0.288i)9-s + (0.241 + 0.417i)10-s + (−0.544 − 0.942i)11-s + (−0.0942 + 0.163i)12-s − 0.0591·13-s + (−0.725 + 0.383i)14-s + 0.339·15-s + (0.283 − 0.490i)16-s + (−0.521 − 0.904i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0256i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.999 - 0.0256i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ 0.999 - 0.0256i)\)

Particular Values

\(L(15)\) \(\approx\) \(1.557157207\)
\(L(\frac12)\) \(\approx\) \(1.557157207\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.39e6 - 4.14e6i)T \)
7 \( 1 + (-1.51e12 - 9.55e11i)T \)
good2 \( 1 + (9.50e3 - 1.64e4i)T + (-2.68e8 - 4.64e8i)T^{2} \)
5 \( 1 + (-4.01e9 + 6.95e9i)T + (-9.31e19 - 1.61e20i)T^{2} \)
11 \( 1 + (6.85e14 + 1.18e15i)T + (-7.93e29 + 1.37e30i)T^{2} \)
13 \( 1 + 8.39e14T + 2.01e32T^{2} \)
17 \( 1 + (3.62e17 + 6.27e17i)T + (-2.40e35 + 4.17e35i)T^{2} \)
19 \( 1 + (6.56e17 - 1.13e18i)T + (-6.06e36 - 1.05e37i)T^{2} \)
23 \( 1 + (2.05e18 - 3.55e18i)T + (-1.54e39 - 2.67e39i)T^{2} \)
29 \( 1 + 2.28e20T + 2.56e42T^{2} \)
31 \( 1 + (-1.18e21 - 2.05e21i)T + (-8.88e42 + 1.53e43i)T^{2} \)
37 \( 1 + (-4.42e22 + 7.66e22i)T + (-1.50e45 - 2.60e45i)T^{2} \)
41 \( 1 + 4.68e23T + 5.89e46T^{2} \)
43 \( 1 - 5.76e23T + 2.34e47T^{2} \)
47 \( 1 + (-1.33e24 + 2.31e24i)T + (-1.54e48 - 2.68e48i)T^{2} \)
53 \( 1 + (1.02e22 + 1.77e22i)T + (-5.04e49 + 8.74e49i)T^{2} \)
59 \( 1 + (2.53e25 + 4.38e25i)T + (-1.13e51 + 1.95e51i)T^{2} \)
61 \( 1 + (-1.62e25 + 2.81e25i)T + (-2.97e51 - 5.15e51i)T^{2} \)
67 \( 1 + (1.97e26 + 3.42e26i)T + (-4.52e52 + 7.82e52i)T^{2} \)
71 \( 1 + 1.16e27T + 4.85e53T^{2} \)
73 \( 1 + (7.80e26 + 1.35e27i)T + (-5.43e53 + 9.41e53i)T^{2} \)
79 \( 1 + (-7.00e26 + 1.21e27i)T + (-5.37e54 - 9.30e54i)T^{2} \)
83 \( 1 - 1.11e28T + 4.50e55T^{2} \)
89 \( 1 + (3.53e27 - 6.12e27i)T + (-1.70e56 - 2.95e56i)T^{2} \)
97 \( 1 - 9.44e28T + 4.13e57T^{2} \)
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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

−12.06524586679698925554687211196, −10.90322178799347981519141615813, −9.178348384377459924614261983114, −8.540814824795739412096627707325, −7.49160883041507873041665018077, −5.88647292371566222179628355034, −4.90752586209394817543547237580, −3.26461865327745726896943531577, −2.09461614253573911628337517326, −0.37420640222801816040877196557, 1.02655539942752818691234551350, 1.95378199100650956153461671896, 2.75058662512145678630746062841, 4.48367307869703476020808690708, 6.11289376673024055993044130652, 7.26717656429316470417540398399, 8.558648751907841136563814148747, 10.02183901530247579723265068934, 10.77542431258398997309265302472, 11.92640496862449379198821121184

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