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Properties

Label 2-6300-15.8-c1-0-15
Degree $2$
Conductor $6300$
Sign $0.927 - 0.374i$
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 + 2.58i·11-s + (−0.585 − 0.585i)13-s + (2.82 + 2.82i)17-s − 2.82i·19-s + (3.82 − 3.82i)23-s − 1.41·29-s + 2.82·31-s + (2.58 − 2.58i)37-s − 6.82i·41-s + (−1.17 − 1.17i)43-s + (1.17 + 1.17i)47-s − 1.00i·49-s + (2.17 − 2.17i)53-s − 4·59-s + ⋯
L(s)  = 1  + (−0.267 + 0.267i)7-s + 0.779i·11-s + (−0.162 − 0.162i)13-s + (0.685 + 0.685i)17-s − 0.648i·19-s + (0.798 − 0.798i)23-s − 0.262·29-s + 0.508·31-s + (0.425 − 0.425i)37-s − 1.06i·41-s + (−0.178 − 0.178i)43-s + (0.170 + 0.170i)47-s − 0.142i·49-s + (0.298 − 0.298i)53-s − 0.520·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.927 - 0.374i$
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: \((2,\ 6300,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.859769313\)
\(L(\frac12)\) \(\approx\) \(1.859769313\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.58iT - 11T^{2} \)
13 \( 1 + (0.585 + 0.585i)T + 13iT^{2} \)
17 \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 + (-3.82 + 3.82i)T - 23iT^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + (-2.58 + 2.58i)T - 37iT^{2} \)
41 \( 1 + 6.82iT - 41T^{2} \)
43 \( 1 + (1.17 + 1.17i)T + 43iT^{2} \)
47 \( 1 + (-1.17 - 1.17i)T + 47iT^{2} \)
53 \( 1 + (-2.17 + 2.17i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 + (5.41 - 5.41i)T - 67iT^{2} \)
71 \( 1 - 0.242iT - 71T^{2} \)
73 \( 1 + (-10.2 - 10.2i)T + 73iT^{2} \)
79 \( 1 - 1.65iT - 79T^{2} \)
83 \( 1 + (-6.82 + 6.82i)T - 83iT^{2} \)
89 \( 1 - 5.17T + 89T^{2} \)
97 \( 1 + (2.24 - 2.24i)T - 97iT^{2} \)
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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

−8.011945280395355083545793900055, −7.37524544828517833328311611227, −6.69853723188675720344569734579, −5.98911274891708471564284754411, −5.19090131718336940611584942520, −4.52416975014747002187853180872, −3.65820739883734864721210823576, −2.77146357323418083992984109813, −1.98958231315904644799824178706, −0.76786541456351912771337114267, 0.67438042989511200365024654148, 1.66124166753060998933903067385, 2.98233991835770123735974889796, 3.35942072173993952807154348698, 4.40857639353131737204305589911, 5.14231025563796303964774450398, 5.93731528278931417904323898541, 6.50797355206326618751970838114, 7.43928563026230911352952778461, 7.85080695210035227556242078657

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