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Properties

Label 2-21-21.20-c29-0-34
Degree $2$
Conductor $21$
Sign $-0.347 + 0.937i$
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  − 3.84e4i·2-s + (8.27e6 − 3.36e5i)3-s − 9.42e8·4-s − 2.26e10·5-s + (−1.29e10 − 3.18e11i)6-s + (6.92e11 − 1.65e12i)7-s + 1.55e13i·8-s + (6.84e13 − 5.57e12i)9-s + 8.71e14i·10-s + 2.70e14i·11-s + (−7.79e15 + 3.17e14i)12-s − 4.56e15i·13-s + (−6.36e16 − 2.66e16i)14-s + (−1.87e17 + 7.62e15i)15-s + 9.35e16·16-s + 1.12e18·17-s + ⋯
L(s)  = 1  − 1.65i·2-s + (0.999 − 0.0406i)3-s − 1.75·4-s − 1.65·5-s + (−0.0674 − 1.65i)6-s + (0.385 − 0.922i)7-s + 1.25i·8-s + (0.996 − 0.0812i)9-s + 2.75i·10-s + 0.214i·11-s + (−1.75 + 0.0713i)12-s − 0.321i·13-s + (−1.53 − 0.640i)14-s + (−1.65 + 0.0674i)15-s + 0.324·16-s + 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.347 + 0.937i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ -0.347 + 0.937i)\)

Particular Values

\(L(15)\) \(\approx\) \(2.239865675\)
\(L(\frac12)\) \(\approx\) \(2.239865675\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.27e6 + 3.36e5i)T \)
7 \( 1 + (-6.92e11 + 1.65e12i)T \)
good2 \( 1 + 3.84e4iT - 5.36e8T^{2} \)
5 \( 1 + 2.26e10T + 1.86e20T^{2} \)
11 \( 1 - 2.70e14iT - 1.58e30T^{2} \)
13 \( 1 + 4.56e15iT - 2.01e32T^{2} \)
17 \( 1 - 1.12e18T + 4.81e35T^{2} \)
19 \( 1 - 5.21e18iT - 1.21e37T^{2} \)
23 \( 1 - 7.10e19iT - 3.09e39T^{2} \)
29 \( 1 + 4.93e20iT - 2.56e42T^{2} \)
31 \( 1 - 4.68e21iT - 1.77e43T^{2} \)
37 \( 1 - 8.67e22T + 3.00e45T^{2} \)
41 \( 1 - 7.90e22T + 5.89e46T^{2} \)
43 \( 1 + 3.53e23T + 2.34e47T^{2} \)
47 \( 1 + 1.01e24T + 3.09e48T^{2} \)
53 \( 1 - 1.01e25iT - 1.00e50T^{2} \)
59 \( 1 + 4.20e25T + 2.26e51T^{2} \)
61 \( 1 - 1.70e25iT - 5.95e51T^{2} \)
67 \( 1 - 3.45e26T + 9.04e52T^{2} \)
71 \( 1 + 5.62e26iT - 4.85e53T^{2} \)
73 \( 1 + 1.75e27iT - 1.08e54T^{2} \)
79 \( 1 - 3.30e27T + 1.07e55T^{2} \)
83 \( 1 + 1.10e27T + 4.50e55T^{2} \)
89 \( 1 - 1.29e28T + 3.40e56T^{2} \)
97 \( 1 + 1.79e28iT - 4.13e57T^{2} \)
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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

−11.83530369303534312314219777087, −10.63309255485442935097519559130, −9.685097281476691876631698493396, −8.089122094118199299227834475521, −7.56311854795109771390794513623, −4.59211785876042064318138901244, −3.60925360752473432925398962548, −3.27703700575316443224386508164, −1.55665595871182324420975756133, −0.819569249395172362234503639036, 0.60854342552194292039422944881, 2.73373778990051697356202322061, 4.07720917083595915204812999358, 5.03427997732442604986971240289, 6.69697280271149377707407301741, 7.83441505198617637774933611047, 8.264146089458317285605141660202, 9.304380222992624425518050662779, 11.48227669090167321750839165472, 12.84359279886450022825139313138

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