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Properties

Label 2-21-21.5-c3-0-2
Degree $2$
Conductor $21$
Sign $0.745 - 0.666i$
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 + (−5.18 − 0.403i)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 + (26.6 + 4.17i)9-s + (45.6 − 26.3i)10-s + (−15.5 + 8.95i)11-s + (−28.3 − 59.3i)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 + (−0.996 − 0.0775i)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 + (0.987 + 0.154i)9-s + (1.44 − 0.833i)10-s + (−0.425 + 0.245i)11-s + (−0.682 − 1.42i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.745 - 0.666i$
Analytic conductor: \(1.23904\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3/2),\ 0.745 - 0.666i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.49812 + 0.572125i\)
\(L(\frac12)\) \(\approx\) \(1.49812 + 0.572125i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.18 + 0.403i)T \)
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

−17.12285811989482312866775149325, −16.43312150669433945142720621722, −15.50813244863604899111714139294, −13.63175911933513274707031154014, −12.93143838522929161155735156675, −11.82375683793024379831610736942, −9.655413092238734279361635725458, −6.99666113566125425994567912339, −5.80657903662348091305374854425, −4.48343967413805275217402009652, 3.13568813027488427376538127440, 5.36456384693872534346677849291, 6.54424658556538739141496036566, 10.22620336085214698463333554533, 10.93148059011702834507792713699, 12.50656818548939246774629338976, 13.22509675772147925026938723235, 14.75919421633463118662900277827, 15.96732523824709959393649896628, 17.68105062665201773716056285994

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