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Properties

Label 2-21-21.20-c5-0-7
Degree $2$
Conductor $21$
Sign $0.617 + 0.786i$
Analytic cond. $3.36806$
Root an. cond. $1.83522$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.57i·2-s + (13.5 + 7.74i)3-s + 0.942·4-s + 46.2·5-s + (43.1 − 75.3i)6-s + (−120. − 48.7i)7-s − 183. i·8-s + (123. + 209. i)9-s − 257. i·10-s − 285. i·11-s + (12.7 + 7.29i)12-s + 1.14e3i·13-s + (−271. + 669. i)14-s + (625. + 357. i)15-s − 992.·16-s − 967.·17-s + ⋯
L(s)  = 1  − 0.985i·2-s + (0.867 + 0.496i)3-s + 0.0294·4-s + 0.826·5-s + (0.489 − 0.854i)6-s + (−0.926 − 0.375i)7-s − 1.01i·8-s + (0.506 + 0.862i)9-s − 0.814i·10-s − 0.711i·11-s + (0.0255 + 0.0146i)12-s + 1.87i·13-s + (−0.370 + 0.913i)14-s + (0.717 + 0.410i)15-s − 0.969·16-s − 0.812·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ 0.617 + 0.786i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.78157 - 0.866177i\)
\(L(\frac12)\) \(\approx\) \(1.78157 - 0.866177i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.5 - 7.74i)T \)
7 \( 1 + (120. + 48.7i)T \)
good2 \( 1 + 5.57iT - 32T^{2} \)
5 \( 1 - 46.2T + 3.12e3T^{2} \)
11 \( 1 + 285. iT - 1.61e5T^{2} \)
13 \( 1 - 1.14e3iT - 3.71e5T^{2} \)
17 \( 1 + 967.T + 1.41e6T^{2} \)
19 \( 1 + 112. iT - 2.47e6T^{2} \)
23 \( 1 - 1.26e3iT - 6.43e6T^{2} \)
29 \( 1 + 598. iT - 2.05e7T^{2} \)
31 \( 1 + 2.46e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.15e3T + 6.93e7T^{2} \)
41 \( 1 + 1.64e4T + 1.15e8T^{2} \)
43 \( 1 - 5.10e3T + 1.47e8T^{2} \)
47 \( 1 - 1.77e4T + 2.29e8T^{2} \)
53 \( 1 + 3.00e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.99e4T + 7.14e8T^{2} \)
61 \( 1 + 3.69e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.46e4T + 1.35e9T^{2} \)
71 \( 1 + 4.36e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.17e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.97e4T + 3.07e9T^{2} \)
83 \( 1 - 2.29e4T + 3.93e9T^{2} \)
89 \( 1 + 6.36e4T + 5.58e9T^{2} \)
97 \( 1 - 1.12e5iT - 8.58e9T^{2} \)
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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

−16.73363789492459460873070866158, −15.74358586723922572439762867067, −13.93109720957562434173947457731, −13.19539163480658201636433765362, −11.36315078340447797401642068937, −9.986274588609924366677329447547, −9.163377765952341475427068646281, −6.69635339135672385365193646548, −3.82694600976095372123911756961, −2.14782822407711496181097665206, 2.55823905972264835698155094103, 5.85408042800250985365412143575, 7.13410586883126512181144588964, 8.604315668678760860259064970267, 10.09026014034889504842671164896, 12.54136115132201333822132880653, 13.57561666316542310741766432709, 14.98472719671611726970226825349, 15.68808496970837239525455747012, 17.34841421336455846135677659951

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