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Properties

Label 2-6300-21.20-c1-0-13
Degree $2$
Conductor $6300$
Sign $0.479 - 0.877i$
Analytic cond. $50.3057$
Root an. cond. $7.09265$
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  + (−1.16 − 2.37i)7-s + 3.74i·11-s + 0.841i·13-s + 3.36·17-s − 4.55i·19-s + 7.64i·23-s − 1.41i·29-s − 0.979i·31-s − 2.32·37-s − 10.3·41-s + 10.8·43-s + 7.91·47-s + (−4.29 + 5.53i)49-s − 4.35i·53-s − 1.38·59-s + ⋯
L(s)  = 1  + (−0.439 − 0.898i)7-s + 1.12i·11-s + 0.233i·13-s + 0.814·17-s − 1.04i·19-s + 1.59i·23-s − 0.262i·29-s − 0.175i·31-s − 0.382·37-s − 1.61·41-s + 1.64·43-s + 1.15·47-s + (−0.613 + 0.790i)49-s − 0.598i·53-s − 0.180·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.479 - 0.877i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (3401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.479 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.471377581\)
\(L(\frac12)\) \(\approx\) \(1.471377581\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 + (1.16 + 2.37i)T \)
good11 \( 1 - 3.74iT - 11T^{2} \)
13 \( 1 - 0.841iT - 13T^{2} \)
17 \( 1 - 3.36T + 17T^{2} \)
19 \( 1 + 4.55iT - 19T^{2} \)
23 \( 1 - 7.64iT - 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 + 0.979iT - 31T^{2} \)
37 \( 1 + 2.32T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 7.91T + 47T^{2} \)
53 \( 1 + 4.35iT - 53T^{2} \)
59 \( 1 + 1.38T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 - 3.74iT - 71T^{2} \)
73 \( 1 - 8.66iT - 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 3.14T + 83T^{2} \)
89 \( 1 - 3.91T + 89T^{2} \)
97 \( 1 - 14.8iT - 97T^{2} \)
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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

−7.955962117049805499943863392645, −7.24835595737797563787248715573, −7.06604593266469171925598200013, −6.05940252228383305682091365743, −5.26478920595629574051539427465, −4.48481276400136821991099977250, −3.80244643874880581675794526648, −3.00436063258152041711286659922, −1.93909416365070199935922542513, −0.942545319618243046784027641019, 0.43714765974186524413431182954, 1.67746644818602910889721331825, 2.80851351314830006293650210174, 3.28505200804742661181677116230, 4.23644823448312011179393873433, 5.22951747099895376651965139271, 5.92219746503583231759717551798, 6.23248978994636300641937163008, 7.26161522353553082681935332788, 8.009769995235847192511715165053

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