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Properties

Label 2-21-7.6-c32-0-40
Degree $2$
Conductor $21$
Sign $-0.352 - 0.935i$
Analytic cond. $136.219$
Root an. cond. $11.6713$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.95e4·2-s − 2.48e7i·3-s − 2.73e9·4-s − 1.96e11i·5-s − 9.81e11i·6-s + (3.11e13 − 1.17e13i)7-s − 2.77e14·8-s − 6.17e14·9-s − 7.76e15i·10-s − 2.70e16·11-s + 6.79e16i·12-s − 4.21e17i·13-s + (1.22e18 − 4.62e17i)14-s − 4.88e18·15-s + 7.75e17·16-s − 6.36e19i·17-s + ⋯
L(s)  = 1  + 0.602·2-s − 0.577i·3-s − 0.636·4-s − 1.28i·5-s − 0.348i·6-s + (0.935 − 0.352i)7-s − 0.986·8-s − 0.333·9-s − 0.776i·10-s − 0.588·11-s + 0.367i·12-s − 0.634i·13-s + (0.564 − 0.212i)14-s − 0.743·15-s + 0.0420·16-s − 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.352 - 0.935i$
Analytic conductor: \(136.219\)
Root analytic conductor: \(11.6713\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :16),\ -0.352 - 0.935i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.250096290\)
\(L(\frac12)\) \(\approx\) \(1.250096290\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 2.48e7iT \)
7 \( 1 + (-3.11e13 + 1.17e13i)T \)
good2 \( 1 - 3.95e4T + 4.29e9T^{2} \)
5 \( 1 + 1.96e11iT - 2.32e22T^{2} \)
11 \( 1 + 2.70e16T + 2.11e33T^{2} \)
13 \( 1 + 4.21e17iT - 4.42e35T^{2} \)
17 \( 1 + 6.36e19iT - 2.36e39T^{2} \)
19 \( 1 + 8.07e19iT - 8.31e40T^{2} \)
23 \( 1 + 1.49e21T + 3.76e43T^{2} \)
29 \( 1 + 4.55e23T + 6.26e46T^{2} \)
31 \( 1 + 1.03e24iT - 5.29e47T^{2} \)
37 \( 1 + 9.28e24T + 1.52e50T^{2} \)
41 \( 1 + 4.65e25iT - 4.06e51T^{2} \)
43 \( 1 - 1.48e26T + 1.86e52T^{2} \)
47 \( 1 + 5.87e26iT - 3.21e53T^{2} \)
53 \( 1 + 4.18e27T + 1.50e55T^{2} \)
59 \( 1 + 6.45e27iT - 4.64e56T^{2} \)
61 \( 1 - 1.60e28iT - 1.35e57T^{2} \)
67 \( 1 - 3.22e29T + 2.71e58T^{2} \)
71 \( 1 + 1.22e28T + 1.73e59T^{2} \)
73 \( 1 - 7.85e29iT - 4.22e59T^{2} \)
79 \( 1 - 8.54e29T + 5.29e60T^{2} \)
83 \( 1 - 2.29e30iT - 2.57e61T^{2} \)
89 \( 1 - 2.92e31iT - 2.40e62T^{2} \)
97 \( 1 + 3.19e31iT - 3.77e63T^{2} \)
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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

−11.32178467409901280191140273197, −9.500827554715601416215210804206, −8.449107529901224675711262610654, −7.51017479268504533983544968484, −5.51486107741660994004767129143, −5.02982081412238140701082989899, −3.87697867014708328966430609495, −2.26455914724374562347522243495, −0.859106594473690382727409926693, −0.25104868329848225464523072325, 1.83389707103053838454344642559, 3.12182473625366885645266923245, 4.06229020362944170967397660530, 5.18692912334575047256945259202, 6.22875599740239271720879248975, 7.84539733089023294920918581605, 9.056588452536587463948012990662, 10.41063354187486822232772629394, 11.28900128084669774465261921940, 12.66038452392101616386607901916

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