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Properties

Label 2-21-7.4-c3-0-0
Degree $2$
Conductor $21$
Sign $0.386 - 0.922i$
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  + (1.5 + 2.59i)2-s + (−1.5 + 2.59i)3-s + (−0.5 + 0.866i)4-s + (1.5 + 2.59i)5-s − 9·6-s + (−3.5 − 18.1i)7-s + 21·8-s + (−4.5 − 7.79i)9-s + (−4.5 + 7.79i)10-s + (7.5 − 12.9i)11-s + (−1.50 − 2.59i)12-s − 64·13-s + (42 − 36.3i)14-s − 9·15-s + (35.5 + 61.4i)16-s + (−42 + 72.7i)17-s + ⋯
L(s)  = 1  + (0.530 + 0.918i)2-s + (−0.288 + 0.499i)3-s + (−0.0625 + 0.108i)4-s + (0.134 + 0.232i)5-s − 0.612·6-s + (−0.188 − 0.981i)7-s + 0.928·8-s + (−0.166 − 0.288i)9-s + (−0.142 + 0.246i)10-s + (0.205 − 0.356i)11-s + (−0.0360 − 0.0625i)12-s − 1.36·13-s + (0.801 − 0.694i)14-s − 0.154·15-s + (0.554 + 0.960i)16-s + (−0.599 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.23904\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.10777 + 0.736870i\)
\(L(\frac12)\) \(\approx\) \(1.10777 + 0.736870i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 + (3.5 + 18.1i)T \)
good2 \( 1 + (-1.5 - 2.59i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-7.5 + 12.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 64T + 2.19e3T^{2} \)
17 \( 1 + (42 - 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-8 - 13.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-42 - 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 297T + 2.43e4T^{2} \)
31 \( 1 + (-126.5 + 219. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-158 - 273. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 360T + 6.89e4T^{2} \)
43 \( 1 - 26T + 7.95e4T^{2} \)
47 \( 1 + (-15 - 25.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (181.5 - 314. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-7.5 + 12.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-59 - 102. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-185 + 320. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 342T + 3.57e5T^{2} \)
73 \( 1 + (181 - 313. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (233.5 + 404. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 477T + 5.71e5T^{2} \)
89 \( 1 + (453 + 784. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 503T + 9.12e5T^{2} \)
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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

−17.23074733756588066166355793207, −16.67477650348304783612437507985, −15.27106905125728587197844724311, −14.40025286648667157123636101921, −13.11785989273716789651633747066, −11.07506946322079037619428719899, −9.873668919387526237337095414518, −7.51666512192279009581180184974, −6.10912565339455725894963758727, −4.37886213680069290113882929938, 2.45162046213851339139222966109, 5.03511410905422182392571092911, 7.24478076920553501392747320947, 9.361730012047625925671515567963, 11.20719166378544860614006310948, 12.28229638658006112017284905068, 13.00932163019732403024923293547, 14.60107513759248829817160017074, 16.30416758363011446028043546281, 17.58386982955827734240686480395

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