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Properties

Label 2-6300-21.5-c1-0-13
Degree $2$
Conductor $6300$
Sign $-0.547 - 0.836i$
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  + (−0.651 + 2.56i)7-s + (−0.362 + 0.209i)11-s + 1.12i·13-s + (2.46 + 4.27i)17-s + (4.27 + 2.46i)19-s + (6.18 + 3.57i)23-s + 4.95i·29-s + (−5.51 + 3.18i)31-s + (4.42 − 7.65i)37-s − 12.1·41-s + 6.05·43-s + (3.85 − 6.67i)47-s + (−6.15 − 3.34i)49-s + (−8.93 + 5.16i)53-s + (−0.569 − 0.986i)59-s + ⋯
L(s)  = 1  + (−0.246 + 0.969i)7-s + (−0.109 + 0.0630i)11-s + 0.312i·13-s + (0.598 + 1.03i)17-s + (0.981 + 0.566i)19-s + (1.28 + 0.744i)23-s + 0.919i·29-s + (−0.991 + 0.572i)31-s + (0.726 − 1.25i)37-s − 1.90·41-s + 0.922·43-s + (0.562 − 0.973i)47-s + (−0.878 − 0.477i)49-s + (−1.22 + 0.708i)53-s + (−0.0741 − 0.128i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.547 - 0.836i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ -0.547 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.591113738\)
\(L(\frac12)\) \(\approx\) \(1.591113738\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.651 - 2.56i)T \)
good11 \( 1 + (0.362 - 0.209i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.12iT - 13T^{2} \)
17 \( 1 + (-2.46 - 4.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.27 - 2.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.18 - 3.57i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.95iT - 29T^{2} \)
31 \( 1 + (5.51 - 3.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.42 + 7.65i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 - 6.05T + 43T^{2} \)
47 \( 1 + (-3.85 + 6.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.93 - 5.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.569 + 0.986i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.9 - 6.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.14 + 8.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.87iT - 71T^{2} \)
73 \( 1 + (-4.00 + 2.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.38 + 4.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (-2.33 + 4.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.49iT - 97T^{2} \)
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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

−8.329409402930069098332083307645, −7.52423093351455000093657219091, −6.95273249877539592806560583527, −6.00580782366936221959246789855, −5.48245587885126822186657810348, −4.89094071245465329181183224466, −3.62320679623099278729876515182, −3.24171567507993308805770780092, −2.10143948613645600212923437929, −1.27330935514490298216110931873, 0.43929784707419004972255588193, 1.24539550230096654439095804187, 2.70321668143087604835479437307, 3.21536037806053794425960196267, 4.18010020752438530817920172632, 4.93992509383852992089627564476, 5.55107278821154051250239307846, 6.58456935319619668323583919006, 7.10012202455430134712994545700, 7.70836877260828052783385866749

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