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Properties

Label 2-6300-15.8-c1-0-24
Degree $2$
Conductor $6300$
Sign $0.374 + 0.927i$
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  + (0.707 − 0.707i)7-s − 2.16i·11-s + (0.457 + 0.457i)13-s + (0.704 + 0.704i)17-s + 2.08i·19-s + (−0.819 + 0.819i)23-s − 5.75·29-s − 4.16·31-s + (7.23 − 7.23i)37-s − 4.82i·41-s + (3.98 + 3.98i)43-s + (3.24 + 3.24i)47-s − 1.00i·49-s + (7.01 − 7.01i)53-s + 4.69·59-s + ⋯
L(s)  = 1  + (0.267 − 0.267i)7-s − 0.651i·11-s + (0.126 + 0.126i)13-s + (0.170 + 0.170i)17-s + 0.477i·19-s + (−0.170 + 0.170i)23-s − 1.06·29-s − 0.748·31-s + (1.18 − 1.18i)37-s − 0.754i·41-s + (0.607 + 0.607i)43-s + (0.472 + 0.472i)47-s − 0.142i·49-s + (0.963 − 0.963i)53-s + 0.611·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (5993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.748113294\)
\(L(\frac12)\) \(\approx\) \(1.748113294\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 + (-0.457 - 0.457i)T + 13iT^{2} \)
17 \( 1 + (-0.704 - 0.704i)T + 17iT^{2} \)
19 \( 1 - 2.08iT - 19T^{2} \)
23 \( 1 + (0.819 - 0.819i)T - 23iT^{2} \)
29 \( 1 + 5.75T + 29T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 + (-7.23 + 7.23i)T - 37iT^{2} \)
41 \( 1 + 4.82iT - 41T^{2} \)
43 \( 1 + (-3.98 - 3.98i)T + 43iT^{2} \)
47 \( 1 + (-3.24 - 3.24i)T + 47iT^{2} \)
53 \( 1 + (-7.01 + 7.01i)T - 53iT^{2} \)
59 \( 1 - 4.69T + 59T^{2} \)
61 \( 1 - 0.627T + 61T^{2} \)
67 \( 1 + (-8.71 + 8.71i)T - 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-0.228 - 0.228i)T + 73iT^{2} \)
79 \( 1 - 2.49iT - 79T^{2} \)
83 \( 1 + (-2.75 + 2.75i)T - 83iT^{2} \)
89 \( 1 + 15.2T + 89T^{2} \)
97 \( 1 + (5.18 - 5.18i)T - 97iT^{2} \)
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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

−7.80799113176254199854654904199, −7.37219515953313640598549974348, −6.42905403745029994481228484974, −5.74037593188779907285610356358, −5.18640705271676618256566575751, −4.02697950797386645216173788182, −3.69494072305990648667099744305, −2.54589271670421984198234738360, −1.64027981375035847202364308658, −0.50363252613536137606852159737, 0.991380839431887853732232588118, 2.08694365758968320035108130544, 2.81672212845011004314284831049, 3.88266932372186527396351449789, 4.52756952791615012178343622246, 5.39730489257901807499629769478, 5.91904445471751319726895166844, 6.94618883014763038939983141910, 7.35846779018372709791167086638, 8.217003107656108972718512035639

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