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Properties

Label 2-6300-5.4-c1-0-28
Degree $2$
Conductor $6300$
Sign $0.447 + 0.894i$
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  − i·7-s + 11-s + 2i·13-s − 8i·17-s + 2·19-s + i·23-s + 29-s + 6·31-s + 9i·37-s + i·43-s − 6i·47-s − 49-s + 2i·53-s − 6·59-s + 8·61-s + ⋯
L(s)  = 1  − 0.377i·7-s + 0.301·11-s + 0.554i·13-s − 1.94i·17-s + 0.458·19-s + 0.208i·23-s + 0.185·29-s + 1.07·31-s + 1.47i·37-s + 0.152i·43-s − 0.875i·47-s − 0.142·49-s + 0.274i·53-s − 0.781·59-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (6049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.848412785\)
\(L(\frac12)\) \(\approx\) \(1.848412785\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good11 \( 1 - T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 8iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 9iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 7T + 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−7.85402300740634683499642464402, −7.11240221074108072589695786607, −6.68807402918247619527077030385, −5.80251415198006960203902236078, −4.86066726143237848397469325973, −4.47662606159424866129878131770, −3.36056250607840505835607877714, −2.73759464070514296062738072543, −1.57435841845552287555689404896, −0.54702362048965279976334571577, 0.993147858956797702808122568249, 2.00306222542157523385055120376, 2.92521723457689171434467502489, 3.80436166487709745462738680674, 4.45668484684864758960679609174, 5.49488355413082216615212915254, 5.96915945365031783969578428443, 6.67943831844853787327134434182, 7.53009940962773471629759904841, 8.269202740732184609784950497556

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