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Properties

Label 2-21-21.17-c3-0-1
Degree $2$
Conductor $21$
Sign $0.571 - 0.820i$
Analytic cond. $1.23904$
Root an. cond. $1.11312$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.24 + 1.29i)2-s + (5.19 − 0.00519i)3-s + (−0.652 + 1.13i)4-s + (8.05 + 13.9i)5-s + (−11.6 + 6.73i)6-s + (−5.67 − 17.6i)7-s − 24.0i·8-s + (26.9 − 0.0539i)9-s + (−36.1 − 20.8i)10-s + (−30.8 − 17.7i)11-s + (−3.38 + 5.87i)12-s − 7.40i·13-s + (35.5 + 32.1i)14-s + (41.9 + 72.4i)15-s + (25.9 + 44.9i)16-s + (14.4 − 25.0i)17-s + ⋯
L(s)  = 1  + (−0.792 + 0.457i)2-s + (0.999 − 0.000999i)3-s + (−0.0815 + 0.141i)4-s + (0.720 + 1.24i)5-s + (−0.791 + 0.458i)6-s + (−0.306 − 0.951i)7-s − 1.06i·8-s + (0.999 − 0.00199i)9-s + (−1.14 − 0.659i)10-s + (−0.845 − 0.487i)11-s + (−0.0814 + 0.141i)12-s − 0.158i·13-s + (0.678 + 0.613i)14-s + (0.722 + 1.24i)15-s + (0.405 + 0.701i)16-s + (0.206 − 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.571 - 0.820i$
Analytic conductor: \(1.23904\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3/2),\ 0.571 - 0.820i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.879722 + 0.459678i\)
\(L(\frac12)\) \(\approx\) \(0.879722 + 0.459678i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.19 + 0.00519i)T \)
7 \( 1 + (5.67 + 17.6i)T \)
good2 \( 1 + (2.24 - 1.29i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-8.05 - 13.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (30.8 + 17.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 7.40iT - 2.19e3T^{2} \)
17 \( 1 + (-14.4 + 25.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-30.4 + 17.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (48.0 - 27.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 68.1iT - 2.43e4T^{2} \)
31 \( 1 + (154. + 89.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-116. - 202. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 370.T + 6.89e4T^{2} \)
43 \( 1 + 187.T + 7.95e4T^{2} \)
47 \( 1 + (-87.3 - 151. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-235. - 136. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-48.4 + 83.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (333. - 192. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-509. + 881. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 125. iT - 3.57e5T^{2} \)
73 \( 1 + (-195. - 112. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-532. - 921. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 601.T + 5.71e5T^{2} \)
89 \( 1 + (752. + 1.30e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 327. iT - 9.12e5T^{2} \)
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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

−18.17659151194507410668843378338, −16.71539984187909563988790317963, −15.42728638273982691569636700450, −13.99125948036571606205368232998, −13.21046404864218772559064619681, −10.49240693425603420725127466931, −9.603473199862205835369872456861, −7.901635756424367680764374988292, −6.85122863260842460225129262524, −3.26619686304507911137097675648, 1.99943855337412098492483906890, 5.28992533284485955674918591454, 8.271311943569577676205305630327, 9.205073230594713557615201222028, 10.07919610848421024578917458021, 12.36924307238351549079117232921, 13.48894451351061046897440295738, 14.93547090427569340760658716496, 16.33759162674889399983255825853, 17.89322266420102161599692600276

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