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Properties

Label 2-4200-5.4-c1-0-47
Degree $2$
Conductor $4200$
Sign $-0.894 - 0.447i$
Analytic cond. $33.5371$
Root an. cond. $5.79112$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − i·7-s − 9-s + 2i·13-s + 6i·17-s + 4·19-s − 21-s + 4i·23-s + i·27-s − 6·29-s − 8·31-s − 10i·37-s + 2·39-s − 10·41-s − 12i·43-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.554i·13-s + 1.45i·17-s + 0.917·19-s − 0.218·21-s + 0.834i·23-s + 0.192i·27-s − 1.11·29-s − 1.43·31-s − 1.64i·37-s + 0.320·39-s − 1.56·41-s − 1.82i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(33.5371\)
Root analytic conductor: \(5.79112\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4200} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 4200,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−7.79276907826150006981993084253, −7.30429705951522914043457123532, −6.67227472830029634070209693639, −5.68879767997465844204741802143, −5.25933633419754947274728934243, −3.84211254162130177098868463801, −3.59148924990920761356962535705, −2.09673938451124290436619254542, −1.49648659738593880773446478805, 0, 1.47736893272648551878602418355, 2.85829013442355309935270011728, 3.22932194731174226854815383465, 4.46479175726322757361838560151, 5.06685047426033120465002119791, 5.73241181607972691553633770643, 6.59445701614279418311597421393, 7.47939955074574041020294383822, 8.064861520757445797939535823241

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