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Properties

Label 2-21-7.4-c5-0-1
Degree $2$
Conductor $21$
Sign $-0.701 - 0.712i$
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  + (4.59 + 7.95i)2-s + (−4.5 + 7.79i)3-s + (−26.1 + 45.2i)4-s + (−11.0 − 19.1i)5-s − 82.6·6-s + (126. + 26.6i)7-s − 186.·8-s + (−40.5 − 70.1i)9-s + (101. − 175. i)10-s + (−208. + 360. i)11-s + (−235. − 407. i)12-s + 797.·13-s + (370. + 1.13e3i)14-s + 198.·15-s + (−18.6 − 32.3i)16-s + (687. − 1.19e3i)17-s + ⋯
L(s)  = 1  + (0.811 + 1.40i)2-s + (−0.288 + 0.499i)3-s + (−0.817 + 1.41i)4-s + (−0.197 − 0.341i)5-s − 0.937·6-s + (0.978 + 0.205i)7-s − 1.02·8-s + (−0.166 − 0.288i)9-s + (0.320 − 0.554i)10-s + (−0.519 + 0.899i)11-s + (−0.471 − 0.817i)12-s + 1.30·13-s + (0.505 + 1.54i)14-s + 0.227·15-s + (−0.0182 − 0.0315i)16-s + (0.577 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(3.36806\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :5/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.739914 + 1.76562i\)
\(L(\frac12)\) \(\approx\) \(0.739914 + 1.76562i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 + (-126. - 26.6i)T \)
good2 \( 1 + (-4.59 - 7.95i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (11.0 + 19.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (208. - 360. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 797.T + 3.71e5T^{2} \)
17 \( 1 + (-687. + 1.19e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.15e3 + 2.00e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-477. - 827. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 7.03e3T + 2.05e7T^{2} \)
31 \( 1 + (630. - 1.09e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (4.88e3 + 8.46e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 5.40e3T + 1.15e8T^{2} \)
43 \( 1 - 1.96e4T + 1.47e8T^{2} \)
47 \( 1 + (1.02e3 + 1.78e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (9.01e3 - 1.56e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (3.71e3 - 6.43e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.74e3 + 3.02e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (7.92e3 - 1.37e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 5.81e4T + 1.80e9T^{2} \)
73 \( 1 + (1.95e4 - 3.38e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.88e3 + 8.45e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 7.03e4T + 3.93e9T^{2} \)
89 \( 1 + (7.21e4 + 1.24e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 7.93e4T + 8.58e9T^{2} \)
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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

−17.19439882985066342462652274103, −15.95585913075066343180120649581, −15.26540180598407412080246546056, −14.08777643776746820189710207587, −12.72163279399230164755742844272, −11.05765279897405870925508428272, −8.792674763018713267488283483167, −7.33858803218183208350367612115, −5.50981562690680467665656021301, −4.40540228070355111938452123360, 1.50596623695673060604327933545, 3.68650261413003741380346724003, 5.68612600372883597512352142615, 8.148839507591337179809388701754, 10.69063636377348519966075350535, 11.19595284008474540462917354504, 12.59742753673654916291021060778, 13.66351218527995835355671435007, 14.77845794790042905049315471082, 16.81520701392112863019080952298

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