Properties

Label 2-6300-5.4-c1-0-5
Degree $2$
Conductor $6300$
Sign $-0.894 - 0.447i$
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  + i·7-s + 4i·13-s + 6i·17-s − 2·19-s − 6i·23-s + 2·31-s + 2i·37-s + 6·41-s + 4i·43-s − 49-s + 6i·53-s − 12·59-s − 10·61-s − 4i·67-s + 12·71-s + ⋯
L(s)  = 1  + 0.377i·7-s + 1.10i·13-s + 1.45i·17-s − 0.458·19-s − 1.25i·23-s + 0.359·31-s + 0.328i·37-s + 0.937·41-s + 0.609i·43-s − 0.142·49-s + 0.824i·53-s − 1.56·59-s − 1.28·61-s − 0.488i·67-s + 1.42·71-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (6049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9022438632\)
\(L(\frac12)\) \(\approx\) \(0.9022438632\)
\(L(\frac{3}{2})\) 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 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.281716469057777371140307637039, −7.80586011148143379981332587606, −6.68204145814409268198327771423, −6.35704843202827958378729047479, −5.61639308039263914779074647007, −4.48993798539236579646064759631, −4.21232326030601155531998486483, −3.07300354767038319654858062063, −2.20369118026938981591620748863, −1.36199919330163688168975755910, 0.22978787488470587200278762617, 1.28261505809220118755366957129, 2.50806991864824026005132246761, 3.22078179842749272114688757198, 4.05596516642667854241614003148, 4.95553524026764991462212641778, 5.52323240729834669004304236979, 6.32443826020287419071147102847, 7.20046675861279683197960201975, 7.64645197101774600023572621347

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