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Properties

Label 2-6300-15.2-c1-0-25
Degree $2$
Conductor $6300$
Sign $0.749 + 0.662i$
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 − 3.59i·11-s + (3.79 − 3.79i)13-s + (1.19 − 1.19i)17-s + 7.28i·19-s + (−0.0711 − 0.0711i)23-s − 5.45·29-s + 9.74·31-s + (−1.96 − 1.96i)37-s + 3.10i·41-s + (2.72 − 2.72i)43-s + (7.38 − 7.38i)47-s + 1.00i·49-s + (8.55 + 8.55i)53-s − 4.94·59-s + ⋯
L(s)  = 1  + (0.267 + 0.267i)7-s − 1.08i·11-s + (1.05 − 1.05i)13-s + (0.290 − 0.290i)17-s + 1.67i·19-s + (−0.0148 − 0.0148i)23-s − 1.01·29-s + 1.74·31-s + (−0.322 − 0.322i)37-s + 0.484i·41-s + (0.415 − 0.415i)43-s + (1.07 − 1.07i)47-s + 0.142i·49-s + (1.17 + 1.17i)53-s − 0.643·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.172028140\)
\(L(\frac12)\) \(\approx\) \(2.172028140\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.59iT - 11T^{2} \)
13 \( 1 + (-3.79 + 3.79i)T - 13iT^{2} \)
17 \( 1 + (-1.19 + 1.19i)T - 17iT^{2} \)
19 \( 1 - 7.28iT - 19T^{2} \)
23 \( 1 + (0.0711 + 0.0711i)T + 23iT^{2} \)
29 \( 1 + 5.45T + 29T^{2} \)
31 \( 1 - 9.74T + 31T^{2} \)
37 \( 1 + (1.96 + 1.96i)T + 37iT^{2} \)
41 \( 1 - 3.10iT - 41T^{2} \)
43 \( 1 + (-2.72 + 2.72i)T - 43iT^{2} \)
47 \( 1 + (-7.38 + 7.38i)T - 47iT^{2} \)
53 \( 1 + (-8.55 - 8.55i)T + 53iT^{2} \)
59 \( 1 + 4.94T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + (1.00 + 1.00i)T + 67iT^{2} \)
71 \( 1 + 7.09iT - 71T^{2} \)
73 \( 1 + (-1.53 + 1.53i)T - 73iT^{2} \)
79 \( 1 - 7.98iT - 79T^{2} \)
83 \( 1 + (7.99 + 7.99i)T + 83iT^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (0.0634 + 0.0634i)T + 97iT^{2} \)
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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

−7.987044415404058303044844135579, −7.47988127290728957802240863222, −6.26959936593638905551455913862, −5.84856027991560259129850563132, −5.34325977872042890040214582977, −4.19052490335283351524589139030, −3.48567360774016065902643755595, −2.81622645749827237523908116929, −1.60069912109769308468867388249, −0.67193498299180137786008389138, 0.970743988461337409622338233709, 1.91309989649562311583730820777, 2.78730946581314471433901424201, 3.91455210935122647558780989577, 4.44528460772140669426480732877, 5.12901564630076058106751749819, 6.14733637071480281205470760476, 6.74252543549199468835363330752, 7.36044122170271429578591025866, 8.060213132691261042686316308681

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