Properties

Degree $2$
Conductor $6300$
Sign $0.447 + 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·7-s + 11-s + 2i·13-s − 6·19-s + i·23-s + 29-s − 2·31-s − 7i·37-s + 8·41-s + i·43-s + 2i·47-s − 49-s − 14i·53-s + 10·59-s − 3i·67-s + ⋯
L(s)  = 1  − 0.377i·7-s + 0.301·11-s + 0.554i·13-s − 1.37·19-s + 0.208i·23-s + 0.185·29-s − 0.359·31-s − 1.15i·37-s + 1.24·41-s + 0.152i·43-s + 0.291i·47-s − 0.142·49-s − 1.92i·53-s + 1.30·59-s − 0.366i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.447 + 0.894i$
Motivic weight: \(1\)
Character: $\chi_{6300} (6049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.563705878\)
\(L(\frac12)\) \(\approx\) \(1.563705878\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good11 \( 1 - T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
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

−7.934720996971580467658431262459, −7.12350791899728955739604573591, −6.57740782418473537831566035843, −5.86906162130743002115587692141, −4.99241485170495571128446592611, −4.16033409312504071078889926326, −3.68640903147524051846045871005, −2.49197133748881539672909304268, −1.72636298507664953735269052166, −0.46438957403768265157818910288, 0.920772175312508306128775001377, 2.10422136582708275422063904289, 2.83445810712559253034214697924, 3.82288345560584941836361164037, 4.52020567753394969491021155998, 5.35323659869098991917157244386, 6.10270164826179713246749413678, 6.64725579999132615347491504221, 7.51244722779132455488311938850, 8.218669316908950529251621970975

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