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Properties

Label 2-1421-29.28-c1-0-34
Degree $2$
Conductor $1421$
Sign $0.973 + 0.228i$
Analytic cond. $11.3467$
Root an. cond. $3.36849$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10i·2-s + 2.10i·3-s − 2.41·4-s + 1.82·5-s + 4.41·6-s + 0.870i·8-s − 1.41·9-s − 3.84i·10-s + 5.07i·11-s − 5.07i·12-s + 4.41·13-s + 3.84i·15-s − 2.99·16-s − 2.97i·17-s + 2.97i·18-s + 2.97i·19-s + ⋯
L(s)  = 1  − 1.48i·2-s + 1.21i·3-s − 1.20·4-s + 0.817·5-s + 1.80·6-s + 0.307i·8-s − 0.471·9-s − 1.21i·10-s + 1.52i·11-s − 1.46i·12-s + 1.22·13-s + 0.991i·15-s − 0.749·16-s − 0.720i·17-s + 0.700i·18-s + 0.681i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1421 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1421\)    =    \(7^{2} \cdot 29\)
Sign: $0.973 + 0.228i$
Analytic conductor: \(11.3467\)
Root analytic conductor: \(3.36849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1421} (1275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1421,\ (\ :1/2),\ 0.973 + 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.986927436\)
\(L(\frac12)\) \(\approx\) \(1.986927436\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 + (-5.24 - 1.23i)T \)
good2 \( 1 + 2.10iT - 2T^{2} \)
3 \( 1 - 2.10iT - 3T^{2} \)
5 \( 1 - 1.82T + 5T^{2} \)
11 \( 1 - 5.07iT - 11T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 + 2.97iT - 17T^{2} \)
19 \( 1 - 2.97iT - 19T^{2} \)
23 \( 1 - 0.828T + 23T^{2} \)
31 \( 1 + 7.53iT - 31T^{2} \)
37 \( 1 - 4.71iT - 37T^{2} \)
41 \( 1 - 1.74iT - 41T^{2} \)
43 \( 1 - 2.61iT - 43T^{2} \)
47 \( 1 - 3.84iT - 47T^{2} \)
53 \( 1 + 4.07T + 53T^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 7.41T + 67T^{2} \)
71 \( 1 + 9.41T + 71T^{2} \)
73 \( 1 - 3.48iT - 73T^{2} \)
79 \( 1 - 5.58iT - 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + 14.3iT - 97T^{2} \)
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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

−9.762535392741635652424363431399, −9.383411511344908263965099045452, −8.348988359600655187625940625995, −7.02818415857588300636999652372, −5.97433410282500889901449366442, −4.85196578228210040843971306024, −4.26283567058577212176884683463, −3.41267352451352865729977044098, −2.34973573869288695493648920764, −1.36663901858020011741297658820, 0.907876342682476674237807957293, 2.17496807231226330352357321830, 3.58351439231243982417880165553, 5.04556383940304277776065465027, 6.04952794306471455023340958832, 6.18841927933114433487982575578, 6.97058032233934560513654613335, 7.86013154367950662266945118745, 8.611936945666684710129569548946, 8.912066763050990215270474746226

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