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Properties

Label 2-21-21.5-c25-0-32
Degree $2$
Conductor $21$
Sign $-0.0858 + 0.996i$
Analytic cond. $83.1593$
Root an. cond. $9.11917$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.97e5 + 4.60e5i)3-s + (−1.67e7 − 2.90e7i)4-s + (−3.65e10 + 1.49e9i)7-s + (4.23e11 − 7.33e11i)9-s + (2.67e13 + 1.54e13i)12-s + 3.65e13i·13-s + (−5.62e14 + 9.75e14i)16-s + (4.23e15 + 2.44e15i)19-s + (2.84e16 − 1.80e16i)21-s + (1.49e17 + 2.58e17i)25-s + 7.79e17i·27-s + (6.57e17 + 1.03e18i)28-s + (−2.13e18 + 1.23e18i)31-s − 2.84e19·36-s + (−3.53e19 + 6.13e19i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.5 − 0.866i)4-s + (−0.999 + 0.0407i)7-s + (0.5 − 0.866i)9-s + (0.866 + 0.500i)12-s + 0.434i·13-s + (−0.499 + 0.866i)16-s + (0.438 + 0.253i)19-s + (0.844 − 0.534i)21-s + (0.499 + 0.866i)25-s + 0.999i·27-s + (0.534 + 0.844i)28-s + (−0.485 + 0.280i)31-s − 36-s + (−0.883 + 1.53i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0858 + 0.996i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.0858 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.0858 + 0.996i$
Analytic conductor: \(83.1593\)
Root analytic conductor: \(9.11917\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :25/2),\ -0.0858 + 0.996i)\)

Particular Values

\(L(13)\) \(\approx\) \(0.4535723394\)
\(L(\frac12)\) \(\approx\) \(0.4535723394\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.97e5 - 4.60e5i)T \)
7 \( 1 + (3.65e10 - 1.49e9i)T \)
good2 \( 1 + (1.67e7 + 2.90e7i)T^{2} \)
5 \( 1 + (-1.49e17 - 2.58e17i)T^{2} \)
11 \( 1 + (5.41e25 - 9.38e25i)T^{2} \)
13 \( 1 - 3.65e13iT - 7.05e27T^{2} \)
17 \( 1 + (-2.88e30 + 4.99e30i)T^{2} \)
19 \( 1 + (-4.23e15 - 2.44e15i)T + (4.65e31 + 8.06e31i)T^{2} \)
23 \( 1 + (5.52e33 + 9.56e33i)T^{2} \)
29 \( 1 - 3.63e36T^{2} \)
31 \( 1 + (2.13e18 - 1.23e18i)T + (9.61e36 - 1.66e37i)T^{2} \)
37 \( 1 + (3.53e19 - 6.13e19i)T + (-8.01e38 - 1.38e39i)T^{2} \)
41 \( 1 + 2.08e40T^{2} \)
43 \( 1 - 1.81e20T + 6.86e40T^{2} \)
47 \( 1 + (-3.17e41 - 5.49e41i)T^{2} \)
53 \( 1 + (6.39e42 - 1.10e43i)T^{2} \)
59 \( 1 + (-9.33e43 + 1.61e44i)T^{2} \)
61 \( 1 + (3.16e22 + 1.82e22i)T + (2.14e44 + 3.72e44i)T^{2} \)
67 \( 1 + (-2.78e22 - 4.82e22i)T + (-2.24e45 + 3.88e45i)T^{2} \)
71 \( 1 - 1.91e46T^{2} \)
73 \( 1 + (-1.45e23 + 8.40e22i)T + (1.91e46 - 3.31e46i)T^{2} \)
79 \( 1 + (-5.23e23 + 9.06e23i)T + (-1.37e47 - 2.38e47i)T^{2} \)
83 \( 1 + 9.48e47T^{2} \)
89 \( 1 + (-2.71e48 - 4.70e48i)T^{2} \)
97 \( 1 + 8.34e24iT - 4.66e49T^{2} \)
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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

−12.32958102987916817447629529291, −10.93769588542310267166742520558, −9.909922353528808902216740778752, −9.089860177433482285979751594375, −6.82631240265340800825864339723, −5.81817265586083186642796244257, −4.76258548085961490439840660594, −3.44887611969264203812925026876, −1.38301641550212403699801117561, −0.19330195581018203165882063752, 0.70259947698886766915235226779, 2.57738821313122023299998347074, 3.93243102676817272886013399893, 5.35335217461519355678739971063, 6.70054823345590201061238253468, 7.76415886095022502637581318901, 9.254067331809483018942902086427, 10.66877993286668557148643295734, 12.14520814469825071149309938846, 12.80820630644158990062138361945

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