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Properties

Label 2-21-7.2-c3-0-3
Degree $2$
Conductor $21$
Sign $0.448 + 0.893i$
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  + (2.27 − 3.94i)2-s + (1.5 + 2.59i)3-s + (−6.38 − 11.0i)4-s + (−8.93 + 15.4i)5-s + 13.6·6-s + (2.26 − 18.3i)7-s − 21.6·8-s + (−4.5 + 7.79i)9-s + (40.7 + 70.5i)10-s + (5.69 + 9.86i)11-s + (19.1 − 33.1i)12-s − 13.0·13-s + (−67.3 − 50.7i)14-s − 53.6·15-s + (1.62 − 2.81i)16-s + (−26.6 − 46.1i)17-s + ⋯
L(s)  = 1  + (0.805 − 1.39i)2-s + (0.288 + 0.499i)3-s + (−0.797 − 1.38i)4-s + (−0.799 + 1.38i)5-s + 0.930·6-s + (0.122 − 0.992i)7-s − 0.958·8-s + (−0.166 + 0.288i)9-s + (1.28 + 2.23i)10-s + (0.156 + 0.270i)11-s + (0.460 − 0.797i)12-s − 0.279·13-s + (−1.28 − 0.969i)14-s − 0.922·15-s + (0.0254 − 0.0440i)16-s + (−0.379 − 0.658i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(1.23904\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3/2),\ 0.448 + 0.893i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.27535 - 0.787359i\)
\(L(\frac12)\) \(\approx\) \(1.27535 - 0.787359i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.26 + 18.3i)T \)
good2 \( 1 + (-2.27 + 3.94i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (8.93 - 15.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-5.69 - 9.86i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 13.0T + 2.19e3T^{2} \)
17 \( 1 + (26.6 + 46.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-21.2 + 36.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (76.0 - 131. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 186.T + 2.43e4T^{2} \)
31 \( 1 + (-78.9 - 136. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (1.87 - 3.24i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 39.3T + 6.89e4T^{2} \)
43 \( 1 - 429.T + 7.95e4T^{2} \)
47 \( 1 + (10.5 - 18.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (182. + 316. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-113. - 196. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (325. - 564. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (72.7 + 125. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 368.T + 3.57e5T^{2} \)
73 \( 1 + (304. + 527. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (455. - 788. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 327.T + 5.71e5T^{2} \)
89 \( 1 + (-18.8 + 32.5i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 722.T + 9.12e5T^{2} \)
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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

−17.85938090637667981523434693904, −15.76998630492813507406694678570, −14.44833881085257986104330674284, −13.69630978552099495546249402480, −11.84060259256191149626344280730, −10.90973191375250389061564735008, −9.974220816746377401584436604345, −7.32358574739059563002668662537, −4.35379303846738909285594228489, −3.05831546450298624563908847700, 4.49089376306427944573774469808, 6.04758516490262056537973288899, 7.952594403274095311263756886792, 8.722991757986355289959678706162, 12.10326828635448701727356668854, 12.82057622668740306790035699203, 14.25503095150408340233960736069, 15.46422482729174497083026037399, 16.27073886035681434729160253309, 17.43774137561550892199505646202

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