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Properties

Label 2-21-21.17-c3-0-3
Degree $2$
Conductor $21$
Sign $-0.353 + 0.935i$
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  + (−3.93 + 2.27i)2-s + (−2.24 − 4.68i)3-s + (6.33 − 10.9i)4-s + (−5.80 − 10.0i)5-s + (19.4 + 13.3i)6-s + (−18.4 + 2.09i)7-s + 21.1i·8-s + (−16.9 + 21.0i)9-s + (45.6 + 26.3i)10-s + (15.5 + 8.95i)11-s + (−65.5 − 5.10i)12-s − 62.4i·13-s + (67.6 − 50.0i)14-s + (−34.1 + 49.7i)15-s + (2.48 + 4.30i)16-s + (10.7 − 18.5i)17-s + ⋯
L(s)  = 1  + (−1.39 + 0.803i)2-s + (−0.431 − 0.902i)3-s + (0.791 − 1.37i)4-s + (−0.518 − 0.898i)5-s + (1.32 + 0.909i)6-s + (−0.993 + 0.112i)7-s + 0.936i·8-s + (−0.627 + 0.778i)9-s + (1.44 + 0.833i)10-s + (0.425 + 0.245i)11-s + (−1.57 − 0.122i)12-s − 1.33i·13-s + (1.29 − 0.955i)14-s + (−0.587 + 0.855i)15-s + (0.0388 + 0.0673i)16-s + (0.152 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.353 + 0.935i$
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: \((2,\ 21,\ (\ :3/2),\ -0.353 + 0.935i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.160556 - 0.232316i\)
\(L(\frac12)\) \(\approx\) \(0.160556 - 0.232316i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.24 + 4.68i)T \)
7 \( 1 + (18.4 - 2.09i)T \)
good2 \( 1 + (3.93 - 2.27i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (5.80 + 10.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-15.5 - 8.95i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 62.4iT - 2.19e3T^{2} \)
17 \( 1 + (-10.7 + 18.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-9.50 + 5.48i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (59.8 - 34.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 265. iT - 2.43e4T^{2} \)
31 \( 1 + (-8.85 - 5.11i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (20.8 + 36.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 31.0T + 6.89e4T^{2} \)
43 \( 1 + 224.T + 7.95e4T^{2} \)
47 \( 1 + (81.8 + 141. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-456. - 263. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (205. - 356. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-223. + 129. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (161. - 280. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 45.4iT - 3.57e5T^{2} \)
73 \( 1 + (486. + 281. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (144. + 250. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 448.T + 5.71e5T^{2} \)
89 \( 1 + (280. + 486. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 214. iT - 9.12e5T^{2} \)
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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

−17.28534975875386268454386139840, −16.41436636276251413368248785633, −15.42610772620412333139074999369, −13.18395901570118084144886582583, −12.00611241831087813444493333860, −10.05744008269523393019179300667, −8.557991898623335540987290668631, −7.44370686395010856201929217494, −5.95681458648031482662014012012, −0.49993227456804802842988061035, 3.45078836605789536516177156316, 6.78596493212918899473536583037, 8.909722926072997755232074585116, 10.01358006164936783452050139361, 11.03409899991445826579799865348, 12.01993489823792389850137882039, 14.51201753215478167686753153561, 16.12882113672643134015040914487, 16.84747267888951757217065231592, 18.30592506851247702144117125683

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