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Properties

Label 2-6300-105.59-c1-0-30
Degree $2$
Conductor $6300$
Sign $-0.172 + 0.985i$
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  + (−0.390 + 2.61i)7-s + (−2.17 − 1.25i)11-s − 2.11·13-s + (−3.89 − 2.24i)17-s + (−4.89 + 2.82i)19-s + (3.99 + 6.91i)23-s − 4.97i·29-s + (6.13 + 3.54i)31-s + (8.09 − 4.67i)37-s + 6.23·41-s + 9.47i·43-s + (−9.04 + 5.22i)47-s + (−6.69 − 2.04i)49-s + (3.74 − 6.48i)53-s + (−0.188 + 0.325i)59-s + ⋯
L(s)  = 1  + (−0.147 + 0.989i)7-s + (−0.656 − 0.378i)11-s − 0.585·13-s + (−0.944 − 0.545i)17-s + (−1.12 + 0.648i)19-s + (0.832 + 1.44i)23-s − 0.923i·29-s + (1.10 + 0.636i)31-s + (1.33 − 0.768i)37-s + 0.973·41-s + 1.44i·43-s + (−1.31 + 0.761i)47-s + (−0.956 − 0.292i)49-s + (0.514 − 0.890i)53-s + (−0.0245 + 0.0424i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.172 + 0.985i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ -0.172 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5701258733\)
\(L(\frac12)\) \(\approx\) \(0.5701258733\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.390 - 2.61i)T \)
good11 \( 1 + (2.17 + 1.25i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.11T + 13T^{2} \)
17 \( 1 + (3.89 + 2.24i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.89 - 2.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.99 - 6.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.97iT - 29T^{2} \)
31 \( 1 + (-6.13 - 3.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.09 + 4.67i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.23T + 41T^{2} \)
43 \( 1 - 9.47iT - 43T^{2} \)
47 \( 1 + (9.04 - 5.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.74 + 6.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.188 - 0.325i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.41 - 4.85i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.25 + 1.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.01iT - 71T^{2} \)
73 \( 1 + (-2.46 + 4.26i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.97 + 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.29iT - 83T^{2} \)
89 \( 1 + (8.85 + 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.14T + 97T^{2} \)
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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

−7.898477825531490107231199848186, −7.22027233336999393553693138815, −6.15558907216805821270581043008, −5.94252397138708834322337362824, −4.87437720681076152412992227472, −4.42100897151749326471487094111, −3.11697999373229175163694570172, −2.63256239922046896507141491036, −1.69296619315365649434107340475, −0.16249056313386535554536986130, 0.921644978380745981134598784167, 2.28468699724608690028170635150, 2.82198763432019033472567318176, 4.13348200567514141079343960147, 4.47668266967273587160962081627, 5.22193326323844450628102355508, 6.43489043816944474366252265056, 6.72626656028076704469698462227, 7.46813633096341476915209963263, 8.262212240647071462984667258262

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